High School
SAT
ACT
University
MCAT
LSAT
DAT

Practice: Evaluating limits

Related Topics

Wize University Calculus 1 Textbook > Limits

Continuity

3 Activities

checklist
Mark Yourself Question
  1. Grab a piece of paper and try this problem yourself.
  2. When you're done, check the "I have answered this question" box below.
  3. View the solution and report whether you got it right or wrong.

Practice: Evaluating limits

The following is the graph of the function f(x)f\left(x\right). Determine when the function g(x)=1ln⁡(f(x))g\left(x\right)=\frac{1}{\ln\left(f\left(x\right)\right)} is continuous.




  • f(x)f\left(x\right) is discontinuous at x=0
More Continuity Questions:
Continuity
For what values of aa is the following function continuous on (−∞,∞)(-\infty,\infty).
f(x)={ax3−x2+sin⁡(πx)x<2x2−(2+a)x+2ax−1x≥2f(x)= \begin{cases} ax^3-x^2+\sin(\pi x) & x<2 \\ \frac{x^2-(2+a)x+2a}{x-1} & x\geq 2 \end{cases}
Continuity
For what values of aa is the following function continuous on (−∞,∞)(-\infty,\infty).
f(x)={ax3−x2+sin⁡(πx)x<2x2−(2+a)x+2ax−1x≥2f(x)= \begin{cases} ax^3-x^2+\sin(\pi x) & x<2 \\ \frac{x^2-(2+a)x+2a}{x-1} & x\geq 2 \end{cases}
Continuity
Find AA and BB so that the following function is continuous on R\mathbb{R}:
f(x)={A(1−cos⁡xsin⁡2x)if x<02x2−x+B  if 0≤x≤1x2+2x−3x2−1   if x>1{f(x)=\left\{\begin{array}{ll} \displaystyle A\left(\frac{1-\cos{x}}{\sin^{2}{x}}\right)\quad\text{if}\,x<0 \\ \\ \displaystyle 2x^2-x+B\quad\quad\,\,\text{if}\,0\leq x\leq 1\\ \\ \displaystyle\frac{x^2+2x-3}{x^2-1}\,\,\,\quad\quad\text{if}\,x> 1 \end{array}\right.}
Continuity
Consider the piecewise function
h(x)={Cx−17x<10C+3xx=102C−4x>10h(x) = \begin{cases} \frac{C}{x-17} & x<10 \\ C+3x & x=10 \\ 2C-4 & x>10 \end{cases}
where C is the same real number in each formula.
Continuous Functions
For what values of aa and bb is the following function continuous? Answers are given in the form (a,b)\left(a,b\right).
f(x)={x2−2x+a,x≤0ax+b,0<x<4x+21,4≤xf(x) = \left \{ \begin{array}{ll} x^2 - 2x + a, & x \leq 0\\ ax + b, & 0 < x < 4\\ \sqrt{x + 21}, & 4 \leq x \end{array} \right .
Continuity
For what values of aa is the following function continuous on (−∞,∞)(-\infty,\infty).
f(x)={ax3−x2+sin⁡(πx)x<2x2−(2+a)x+2ax−1x≥2f(x)= \begin{cases} ax^3-x^2+\sin(\pi x) & x<2 \\ \frac{x^2-(2+a)x+2a}{x-1} & x\geq 2 \end{cases}
Continuity
For what values of aa is the following function continuous on (−∞,∞)(-\infty,\infty).
f(x)={ax3−x2+sin⁡(πx)x<2x2−(2+a)x+2ax−1x≥2f(x)= \begin{cases} ax^3-x^2+\sin(\pi x) & x<2 \\ \frac{x^2-(2+a)x+2a}{x-1} & x\geq 2 \end{cases}
Special Limit: Continuity
Evaluate lim⁡x→1+arctan⁡(e(πx−1))\lim_{x\to1^+}\arctan\left(e^{\left(\frac{\pi}{x-1}\right)}\right).
Continuous Functions
For what values of aa and bb is the following function continuous? Answers are given in the form (a,b)(a,b).
f(x)={x2−2x+a,x≤0ax+b,0<x<4x+21,4≤xf(x) = \left \{ \begin{array}{ll} x^2 - 2x + a, & x \leq 0\\ ax + b, & 0 < x < 4\\ \sqrt{x + 21}, & 4 \leq x \end{array} \right .
Continuity
Use the graph below to find
Continuity
Use the graph below to find
Continuity
Determine the values of aa and bb such that f(x)f(x) is continuous, where
f(x)={ax+b,if x≤12b+1,if 1<x≤24ax−13b,if x>2f(x)= \begin{cases} ax+b & , \text{if } x \leq 1\\ 2b +1& , \text{if } 1 <x\le2 \\ 4ax-13b&, \text{if } x>2 \end{cases}
Continuity of Piecewise Functions
Find the values of 𝑎 and 𝑏 that make 𝑓 continuous everywhere
f(x)={x2−2if  x<2axif  2≤x<32x−bif  x≥3f\left(x\right)= \begin{cases} x^2-2 & \text{if }\space x<2\\ ax & \text{if }\space 2\le x<3\\ 2x-b & \text{if }\space x\ge3 \end{cases}
Continuous Functions
For what values of aa and bb is the following function continuous?
f(x)={x2−2x+a,x≤0ax+b,0<x<4x+21,4≤xf(x) = \left \{ \begin{array}{ll} x^2 - 2x + a, & x \leq 0\\ ax + b, & 0 < x < 4\\ \sqrt{x + 21}, & 4 \leq x \end{array} \right .
Special Limit: Continuity
Evaluate lim⁡x→1+arctan⁡(e(πx−1))\lim_{x\to1^+}\arctan\left(e^{\left(\frac{\pi}{x-1}\right)}\right).
Continuity of Piecewise Functions
Practice: Continuity of Piecewise Functions
Find the values of 𝑎 and 𝑏 that make 𝑓 continuous everywhere
f(x)={x2−2if  x<2axif  2≤x<32x−bif  x≥3f\left(x\right)= \begin{cases} x^2-2 & \text{if }\space x<2\\ ax & \text{if }\space 2\le x<3\\ 2x-b & \text{if }\space x\ge3 \end{cases}
Continuous Functions
For what values of aa and bb is the following function continuous?
f(x)={x2−2x+a,x≤0ax+b,0<x<4x+21,4≤xf(x) = \left \{ \begin{array}{ll} x^2 - 2x + a, & x \leq 0\\ ax + b, & 0 < x < 4\\ \sqrt{x + 21}, & 4 \leq x \end{array} \right .
Special Limit: Continuity
Evaluate lim⁡x→1+arctan⁡(e(πx−1))\lim_{x\to1^+}\arctan\left(e^{\left(\frac{\pi}{x-1}\right)}\right).
Continuity
Determine the values of aa and bb such that f(x)f(x) is continuous, where
f(x)={ax+b,if x≤12b+1,if 1<x≤24ax−13b,if x>2f(x)= \begin{cases} ax+b & , \text{if } x \leq 1\\ 2b +1& , \text{if } 1 <x\le2 \\ 4ax-13b&, \text{if } x>2 \end{cases}
Continuity
Determine the values of aa and bb such that f(x)f(x) is continuous, where
f(x)={ax+b,if x≤12b+1,if 1<x≤24ax−13b,if x>2f(x)= \begin{cases} ax+b & , \text{if } x \leq 1\\ 2b +1& , \text{if } 1 <x\le2 \\ 4ax-13b&, \text{if } x>2 \end{cases}
Continuity of Piecewise Functions
Practice: Continuity of Piecewise Functions
On which intervals is the function
f(x)={e+ln⁡(x+2)if  x<−1arcsin⁡x+eif  −1≤x<0ex2+1if  x≥0f\left(x\right)= \begin{cases} e+\ln(x+2) & \text{if }\space x<-1\\ \arcsin x+e &\text{if }\space -1\le x<0\\ e^{x^2+1}&\text{if }\space x\ge0 \end{cases}
Practice: Evaluating limits
Practice: Evaluating limits
The following is the graph of the function f(x)f\left(x\right). Determine when the function g(x)=1ln⁡(f(x))g\left(x\right)=\frac{1}{\ln\left(f\left(x\right)\right)} is continuous.
f(x)f\left(x\right) is discontinuous at x=0
Special Limit: Continuity
Evaluate lim⁡x→1+arctan⁡(e(πx−1))\lim_{x\to1^+}\arctan\left(e^{\left(\frac{\pi}{x-1}\right)}\right).
Continuous Functions
Practice: Continuous Functions
Suppose that f(x)f\left(x\right) and g(x)g\left(x\right) are continuous functions on (−3, 2)\left(-3,\ 2\right). If g(1)=−3g\left(1\right)=-3 and lim⁡x→1(ln⁡(4+g(x))−g(x)ef(x))=1\displaystyle\lim_{x\to1}\left(\ln\left(4+g\left(x\right)\right)-\frac{g\left(x\right)}{e^{^{f\left(x\right)}}}\right)=1, determine the value of f(1)f\left(1\right).
Continuity of Piecewise Functions
Practice: Continuity of Piecewise Functions
Find the values of 𝑎 and 𝑏 that make 𝑓 continuous everywhere
f(x)={x2−2if  x<2axif  2≤x<32x−bif  x≥3f\left(x\right)= \begin{cases} x^2-2 & \text{if }\space x<2\\ ax & \text{if }\space 2\le x<3\\ 2x-b & \text{if }\space x\ge3 \end{cases}
Logarithmic Function & Continuity
The following is the graph of y=f(x)y=f\left(x\right). On what interval is ln⁡(f(x)−1)\ln\left(f\left(x\right)-1\right) continuous?
Continuity of Piecewise Functions
Practice: Continuity of Piecewise Functions
On which intervals is the function
f(x)={e+ln⁡(x+2)if  x<−1arcsin⁡x+eif  −1≤x<0ex2+1if  x≥0f\left(x\right)= \begin{cases} e+\ln(x+2) & \text{if }\space x<-1\\ \arcsin x+e &\text{if }\space -1\le x<0\\ e^{x^2+1}&\text{if }\space x\ge0 \end{cases}
Continuity
Find all the points at which the following function is continuous:
f(x)={x3−8x2−4+1if x≠−24       if x=25       if x=−2f(x)=\left\{\begin{array}{ll} \displaystyle \frac{x^3-8}{x^2-4}+1\quad\text{if}\,x\neq -2 \\ \\ 4\quad\qquad\quad\,\,\,\,\,\,\,\text{if}\,x=2\\ \\ 5\quad\qquad\quad\,\,\,\, \,\,\,\text{if}\,x=-2 \end{array}\right.
Practice: Remove Discontinuity
Q.\textbf{Q.} Consider the function f(x)=cos⁡x−1x\displaystyle f(x)=\frac{\cos{x}-1}{x}. Define a function g(x)g(x) which is equal to ff everywhere where ff is defined, and that is continuous on R\mathbb{R}.
Continuity
Assume a, b∈Ra,\ b \in \mathbb{R} and f(x)f(x) is the function given by
f(x)={ex−1,if  x>1a,if  x=1−x2+b,if  x<1f(x)=\begin{cases} e^{x-1},& \text{if}\;x>1\\ a,&\text{if}\;x=1\\ -x^2+b,&\text{if}\;x<1 \end{cases}
Continuity
Find AA so that the following function is continuous at x=1x=1:
f(x)={2x2−x+A  if  0≤x≤1x2+2x−3x2−1   if  x>1{f(x)=\left\{\begin{array}{ll} \displaystyle 2x^2-x+A\quad\quad\,\,\text{if }\,0\leq x\leq 1\\ \\ \displaystyle\frac{x^2+2x-3}{x^2-1}\,\,\,\quad\quad\text{if }\,x> 1 \end{array}\right.}
Continuity
Find AA and BB so that the following function is continuous on R\mathbb{R}:
f(x)={A(1−cos⁡xsin⁡2x)if x<02x2−x+B  if 0≤x≤1x2+2x−3x2−1   if x>1{f(x)=\left\{\begin{array}{ll} \displaystyle A\left(\frac{1-\cos{x}}{\sin^{2}{x}}\right)\quad\text{if}\,x<0 \\ \\ \displaystyle 2x^2-x+B\quad\quad\,\,\text{if}\,0\leq x\leq 1\\ \\ \displaystyle\frac{x^2+2x-3}{x^2-1}\,\,\,\quad\quad\text{if}\,x> 1 \end{array}\right.}
Q.\textbf{Q.} Consider the function
f(x)={x2−1x+1    if x<−1x2−3if x≥−1{f(x)=\left\{\begin{array}{ll} \displaystyle\dfrac{x^2-1}{x+1}\quad\,\,\,\,\text{if}\,x<-1 \\\text{}\\ x^2-3\quad\quad\text{if}\,x\geq-1 \end{array}\right.}
Is it continuous on the interval (−4,4)(-4,4)?
Consider the function
f(x)={x2−1x+1    if x<−1x2−3if x≥−1{f(x)=\left\{\begin{array}{ll} \displaystyle\dfrac{x^2-1}{x+1}\quad\,\,\,\,\text{if}\,x<-1 \\\text{}\\ x^2-3\quad\quad\text{if}\,x\geq-1 \end{array}\right.}
Is it continuous on the interval (−4,4)(-4,4)?
Consider the function f(x)=cos⁡2x−1x2f(x)=\dfrac{\cos^2{x}-1}{x^2}. Define a function g(x)g(x) which is equal to ff everywhere where ff is defined and that is continuous on all R\mathbb{R}.
Find values of aa and bb such that the following function ff is continuous at x=0x=0 and x=1x=1:
f(x)={a(1−cos⁡2xsin⁡(2x)), if x<02x2−x+b, if 0≤x≤1x2+2x−3x2−1, if x>1f(x)=\begin{cases} a\left(\dfrac{\sqrt{1-\cos^2x}}{\sin(2x)}\right)&,\text{ if }x<0\\\\ 2x^2-x+b&,\text{ if }0\le x\le1\\\\ \dfrac{x^2+2x-3}{x^2-1}&,\text{ if }x>1 \end{cases}
Practice: Continuity of piecewise functions
What do the constants a and b have to be such that the piecewise-define function is continuous at x=−2 ?x = -2\ ?
f(x)={x2+6x+8x+2asin⁡(x+2)b(x+2)if x < −2if x= −2if x > −2f(x)=\begin{cases} \frac{x^2+6x+8}{x+2}\\ a\\ \frac{\sin(x+2)}{b(x+2)} \end{cases}\begin{array}{l} if\ x\ <\ -2\\ if\ x=\ -2\\ if\ x\ >\ -2 \end{array}
Practice: Continuity of piecewise functions
What do the constants a and b have to be such that the piecewise-define function is continuous at x=−2 ?x = -2\ ?
f(x)={x2+6x+8x+2asin⁡(x+2)b(x+2)if x < −2if x= −2if x > −2f(x)=\begin{cases} \frac{x^2+6x+8}{x+2}\\ a\\ \frac{\sin(x+2)}{b(x+2)} \end{cases}\begin{array}{l} if\ x\ <\ -2\\ if\ x=\ -2\\ if\ x\ >\ -2 \end{array}
Practice: Continuity of piecewise functions
What do the constants a and b have to be such that the piecewise-define function is continuous at x=−2 ?x = -2\ ?
f(x)={x2+6x+8x+2asin⁡(x+2)b(x+2)if x < −2if x= −2if x > −2f(x)=\begin{cases} \frac{x^2+6x+8}{x+2}\\ a\\ \frac{\sin(x+2)}{b(x+2)} \end{cases}\begin{array}{l} if\ x\ <\ -2\\ if\ x=\ -2\\ if\ x\ >\ -2 \end{array}
Piecewise Functions
Practice: Piecewise Functions
What value of 'a' would make the following piecewise function continuous?
f(x)={x2+10;x<1−x+7a;x≥1f(x)=\begin{cases} x^2+10;&x<1\\ -x+7a;&x\geq1 \end{cases}
Piecewise Functions
Practice: Piecewise Functions
What value of 'a' would make the following piecewise function continuous?
f(x)={−2x2+7;x<−43x+a;x≥−4f(x)=\begin{cases} -2x^2+7;&x<-4\\ 3x+a;&x\geq-4 \end{cases}
Continuity and Piecewise Functions
What values of aa and bb make the following function continuous?
f(x)={ax2−4;x<−2ax+b;−2≤x≤3ax2+bx+10;x>3f(x)= \begin{cases} ax^2-4;&&x<-2\\\\ ax+b;&&-2\leq{}x\leq3\\\\ ax^2+bx+10;&&x>3 \end{cases}
Continuity
Determine whether the following function is continuous at x=0x = 0 using the definition of continuity. You must full justify your solution.
g(x)={x2+4−2x , x<00 , x≥0\displaystyle g(x) = \begin{cases} \frac{\sqrt{x^2 + 4} - 2}{x} & \text{ , } x < 0 \\ 0 & \text{ , } x \geq 0 \end{cases}
Continuity
Let f(x)f(x) be a continuous function on the open interval (a,b)(a, b). Which of the following four statements are always true.
Select all that apply.
Continuity
Where is f(x)=cos⁡(πx2)1−x2f(x) = \frac{\cos\left(\frac{\pi x}{2} \right)}{\sqrt{1 - x^2}} continuous?
Infinite Limits and Continuity
Given the function f(x)={arctan⁡x+π/2if x≤02xif 0<x≤e1ln⁡xif x>ef(x)=\begin{cases} \arctan x+\pi/2&\text{if }x \le 0\\ \frac{2}{x}&\text{if }0<x\le e\\ \frac{1}{\ln x}&\text{if }x>e \end{cases}, which of the following statements is true about f(x)f(x)?
f(x)={a2x2+1if x<1x+4if x=1−5(a+x)if x>1f\left(x\right)= \begin{cases} a^2x^2+1&if\space x<1\\ x+4 & if \space x=1\\ -5(a+x)&if~x>1 \end{cases}
a) For what value(s) of aa does lim⁡x→1f(x)\displaystyle \lim_{x\to1}f(x) exist?
b) For what value(s) of aa is f(x)f(x) continuous at x=1x=1?
Continuity
Use the graph below to find
Find AA and BB such that the following function is continuous:
f(x)={A(1−cos2xsin⁡2x)if x<02x2−x+B       if 0≤x≤1x2+2x−3x2−1        if x>1.f(x)=\left\{ \begin{array}{ll} \displaystyle A\left(\dfrac{\sqrt{1-cos^2x}}{\sin{2x}}\right)\quad\text{if}\,x<0 \\ 2x^2-x+B\quad\,\,\,\,\,\,\,\text{if}\,0\leq x\leq 1\\ \displaystyle\dfrac{x^2+2x-3}{x^2-1}\quad\,\,\,\,\, \,\,\,\text{if}\,x>1. \end{array} \right.
Find AA and BB such that the following function is continuous:
f(x)={A(1−cos2xsin⁡2x)if x<02x2−x+B       if 0≤x≤1x2+2x−3x2−1        if x>1.f(x)=\left\{ \begin{array}{ll} \displaystyle A\left(\dfrac{\sqrt{1-cos^2x}}{\sin{2x}}\right)\quad\text{if}\,x<0 \\ 2x^2-x+B\quad\,\,\,\,\,\,\,\text{if}\,0\leq x\leq 1\\ \displaystyle\dfrac{x^2+2x-3}{x^2-1}\quad\,\,\,\,\, \,\,\,\text{if}\,x>1. \end{array} \right.
Practice: Continuity of Piecewise Functions
Consider the function
f(x)={ln⁡(x2)+xif  x<−1cos⁡πx2if  −1≤x<12exif  x≥12f\left(x\right)= \begin{cases} \ln(x^2)+x & \text{if }\space x<-1\\ \cos \frac{{\pi}x}{2}&\text{if }\space -1\le x<\frac{1}{2}\\ e^{x}&\text{if }\space x\ge \frac{1}{2} \end{cases}
Limits: Continuity
Find all the points at which the following function is continuous:
f(x)={x3−8x2−4+1 if x≠−2, x≠24 if x=25 if x=−2f(x) = \begin{cases} \frac{x^3-8}{x^2-4}+1 & \text{ if } x\neq -2, \, x \neq 2 \\ 4 & \text{ if } x = 2 \\ 5 & \text{ if } x = -2 \end{cases}
Limits: Continuity
Find A and B so that the following function is continuous at 0 and 1:
f(x)={A(1−cos⁡2xsin⁡2x)x<02x2−x+B0≤x≤1x2+2x−3x2−1x>1f(x) = \begin{cases} A\left(\frac{\sqrt{1-\cos^2 x}}{\sin 2x}\right) & x < 0 \\ 2x^2-x+B & 0 \leq x \leq 1 \\ \frac{x^2+2x-3}{x^2-1} & x > 1 \end{cases}
Enter your answer as A,B. For instance, if you think A=0, B=1, enter 0,1
If f(x)f(x) is continuous at x=−2x=-2 and if lim⁡x→−2(2f(x)−x)2∣f(x)∣2+1=4\lim\limits_{x\rightarrow -2}\frac{(2f(x)-x)^2}{|f(x)|^2+1}=4, then
Continuity
Find all the points at which the following function is continuous:
f(x)={x3−8x2−4+1if x≠−24       if x=25       if x=−2f(x)=\left\{\begin{array}{ll} \displaystyle \frac{x^3-8}{x^2-4}+1\quad\text{if}\,x\neq -2 \\ \\ 4\quad\qquad\quad\,\,\,\,\,\,\,\text{if}\,x=2\\ \\ 5\quad\qquad\quad\,\,\,\, \,\,\,\text{if}\,x=-2 \end{array}\right.
Continuity
Find AA and BB so that the following function is continuous on R\mathbb{R}:
f(x)={A(1−cos⁡xsin⁡2x)if x<02x2−x+B  if 0≤x≤1x2+2x−3x2−1   if x>1{f(x)=\left\{\begin{array}{ll} \displaystyle A\left(\frac{1-\cos{x}}{\sin^{2}{x}}\right)\quad\text{if}\,x<0 \\ \\ \displaystyle 2x^2-x+B\quad\quad\,\,\text{if}\,0\leq x\leq 1\\ \\ \displaystyle\frac{x^2+2x-3}{x^2-1}\,\,\,\quad\quad\text{if}\,x> 1 \end{array}\right.}
Q.\textbf{Q.} Consider the function
f(x)={x2−1x+1    if x<−1x2−3if x≥−1{f(x)=\left\{\begin{array}{ll} \displaystyle\dfrac{x^2-1}{x+1}\quad\,\,\,\,\text{if}\,x<-1 \\\text{}\\ x^2-3\quad\quad\text{if}\,x\geq-1 \end{array}\right.}
Is it continuous on the interval (−4,4)(-4,4)?
Practice: Remove Discontinuity
Q.\textbf{Q.} Consider the function f(x)=cos⁡x−1x\displaystyle f(x)=\frac{\cos{x}-1}{x}. Define a function g(x)g(x) which is equal to ff everywhere where ff is defined, and that is continuous on R\mathbb{R}.
Continuity
Assume a, b∈Ra,\ b \in \mathbb{R} and f(x)f(x) is the function given by
f(x)={ex−1,if  x>1a,if  x=1−x2+b,if  x<1f(x)=\begin{cases} e^{x-1},& \text{if}\;x>1\\ a,&\text{if}\;x=1\\ -x^2+b,&\text{if}\;x<1 \end{cases}
Continuity
Find AA so that the following function is continuous at x=1x=1:
f(x)={2x2−x+A  if  0≤x≤1x2+2x−3x2−1   if  x>1{f(x)=\left\{\begin{array}{ll} \displaystyle 2x^2-x+A\quad\quad\,\,\text{if }\,0\leq x\leq 1\\ \\ \displaystyle\frac{x^2+2x-3}{x^2-1}\,\,\,\quad\quad\text{if }\,x> 1 \end{array}\right.}
Find all number(s) C that make(s) f(x) continuous.
f(x)={x−C1+Cx≤02x2+Cx>0f(x)=\begin{cases}\displaystyle\frac{x-C}{1+C}&x\leq0\\2x^2+C&x>0\end{cases}
Discuss where the function is continuous: f(x)=sin⁡(x+cos⁡x)f(x)=\sin(x+\cos x).
If f(x)f(x) is continuous at x=2x=2 and if lim⁡x→23f(x)+2xf(x)−2=4\lim\limits_{x\rightarrow 2}\frac{3f(x)+2}{xf(x)-2}=4, then
Continuity
Find aa such that f(x)f(x) is continuous at x=1x = 1
f(x)={4x2−3ln⁡xx≥13x3−4axx<1f(x)= \begin{cases} 4x^2 - 3\ln{x} & x \ge 1 \\ 3x^3 - 4ax & x < 1 \end{cases}
Continuity
For what values of aa is the following function continuous on (−∞,∞)(-\infty,\infty).
f(x)={ax3−x2+sin⁡(πx)x<2x2−(2+a)x+2ax−1x≥2f(x)= \begin{cases} ax^3-x^2+\sin(\pi x) & x<2 \\ \frac{x^2-(2+a)x+2a}{x-1} & x\geq 2 \end{cases}
Continuity
Consider the piecewise function
h(x)={Cx−17x<10C+3xx=102C−4x>10h(x) = \begin{cases} \frac{C}{x-17} & x<10 \\ C+3x & x=10 \\ 2C-4 & x>10 \end{cases}
where C is the same real number in each formula.
Continuity
Discuss where the following function is continuous:
h(x)=log⁡(x+1−2)h(x)=\log(\sqrt{x+1}-2)
Consider f(x)=sin⁡2xx\displaystyle f(x)=\frac{\sin{2x}}{x} . Does lim⁡x→0sin⁡2xx\displaystyle \lim_{x\rightarrow 0}\frac{\sin{2x}}{x} exist? Is ff continuous at x=0x=0?
When is the following function continuous: g(x)=x2−1x2−1\displaystyle g(x)=\frac{x^2-1}{x^2-1}?
Where is the following function continuous?
f(x)=ln⁡(x2−1)f(x)=\ln(x^2-1)
Find aa,bb such that f(x)f(x) is continuous everywhere.
f(x)={x2+x+ax>1x−b0≤x≤1x3−4x+5x<0f(x)=\begin{cases}x^2+x+a&x>1\\x-b&0\leq x\leq1\\x^3-4x+5&x<0\end{cases}
🦊 TRICKY! Find A > 0 and B so that the following function is continuous at x = 0 and x = 1:
f(x)={x2tan⁡2Axifx<03x2+2x+Bif0≤x≤13x2−7x−20x2−3x−4ifx>1f(x)=\begin{cases} \frac{x^2}{\tan^2Ax}&\text{if}&x<0\\ 3x^2+2x+B&\text{if}&0\leq x\leq 1\\ \frac{3x^2-7x-20}{x^2-3x-4}&\text{if}&x>1\end{cases}
Continuity
Consider the function f(x)=cos⁡2x−1x2f(x)=\dfrac{\cos^2{x}-1}{x^2}. Define a function g(x)g(x) which is equal to ff everywhere where ff is defined and that is continuous on all R\mathbb{R}.
Practice: Continuity of piecewise functions
What do the constants a and b have to be such that the piecewise-define function is continuous at x=−2 ?x = -2\ ?
f(x)={x2+6x+8x+2asin⁡(x+2)b(x+2)if x < −2if x= −2if x > −2f(x)=\begin{cases} \frac{x^2+6x+8}{x+2}\\ a\\ \frac{\sin(x+2)}{b(x+2)} \end{cases}\begin{array}{l} if\ x\ <\ -2\\ if\ x=\ -2\\ if\ x\ >\ -2 \end{array}
Limits: Continuity of piecewise functions
Identify the type of discontinuity (or state the function is continuous) at x=5,10,15x=5, 10, 15. Is there another point of discontinuity in the domain other than at the stated x-values?
f(x)={x+3x−6x−6+7ln⁡(15−x)sin⁡xif−∞<x≤5if 5<x≤10if10<x<15if15≤x<∞f(x)=\begin{cases} x+3\\ \frac{x-6}{x-6}+7\\ \ln(15-x)\\ \sin x \end{cases}\begin{array}{l} if -\infty < x \leq 5\\ if\ 5 < x \leq 10\\ if 10 < x< 15\\ if 15 \leq x < \infty \end{array}
Find AA and BB such that the following function is continuous:
f(x)={A(1−cos2xsin⁡2x)if x<02x2−x+B       if 0≤x≤1x2+2x−3x2−1        if x>1.f(x)=\left\{ \begin{array}{ll} \displaystyle A\left(\dfrac{\sqrt{1-cos^2x}}{\sin{2x}}\right)\quad\text{if}\,x<0 \\ 2x^2-x+B\quad\,\,\,\,\,\,\,\text{if}\,0\leq x\leq 1\\ \displaystyle\dfrac{x^2+2x-3}{x^2-1}\quad\,\,\,\,\, \,\,\,\text{if}\,x>1. \end{array} \right.
Continuity
Find the value of aa such that the function
f(x)={x2+1if x≤−3ax−1if x>−3f\left(x\right)= \begin{cases} x^2+1&if\space x\le -3\\ ax-1 & if \space x>-3 \end{cases}
is continuous everywhere.
Special Limit: Continuity
Evaluate lim⁡x→1+arctan⁡(e(πx−1))\lim_{x\to1^+}\arctan\left(e^{\left(\frac{\pi}{x-1}\right)}\right).
Continuity of Piecewise Functions
Practice: Continuity of Piecewise Functions
On which intervals is the function
f(x)={e+ln⁡(x+2)if  x<−1arcsin⁡x+eif  −1≤x<0ex2+1if  x≥0f\left(x\right)= \begin{cases} e+\ln(x+2) & \text{if }\space x<-1\\ \arcsin x+e &\text{if }\space -1\le x<0\\ e^{x^2+1}&\text{if }\space x\ge0 \end{cases}
Continuity of Piecewise Functions
Practice: Continuity of Piecewise Functions
Find the values of 𝑎 and 𝑏 that make 𝑓 continuous everywhere
f(x)={x2−2if  x<2axif  2≤x<32x−bif  x≥3f\left(x\right)= \begin{cases} x^2-2 & \text{if }\space x<2\\ ax & \text{if }\space 2\le x<3\\ 2x-b & \text{if }\space x\ge3 \end{cases}
Practice: Evaluating limits
The following is the graph of the function f(x)f\left(x\right), which is defined only defined on [−5,9][-5,9]. Determine when the function g(x)=1ln⁡(f(x))g\left(x\right)=\dfrac{1}{\ln\left(f\left(x\right)\right)} is continuous.
Continuous Functions
Practice: Continuous Functions
Suppose that f(x)f\left(x\right) and g(x)g\left(x\right) are continuous functions on (−3, 2)\left(-3,\ 2\right). If g(1)=−3g\left(1\right)=-3 and lim⁡x→1(ln⁡(4+g(x))−g(x)ef(x))=1\displaystyle\lim_{x\to1}\left(\ln\left(4+g\left(x\right)\right)-\frac{g\left(x\right)}{e^{^{f\left(x\right)}}}\right)=1, determine the value of f(1)f\left(1\right).
Logarithmic Function & Continuity
The following is the graph of y=f(x)y=f\left(x\right). On what interval is ln⁡(f(x)−1)\ln\left(f\left(x\right)-1\right) continuous?
Each question is worth 3 marks.
Continuity
What value of aa makes the following function continuous?
f(x)={2x+4,x≤2ax2+x−2,2<xf(x) = \left \{ \begin{array}{ll} 2^x + 4, & x \leq 2\\ ax^2 + x - 2, & 2 < x \end{array} \right .
Continuous Functions
For what values of aa and bb is the following function continuous?
f(x)={x2−2x+a,x≤0ax+b,0<x<4x+21,4≤xf(x) = \left \{ \begin{array}{ll} x^2 - 2x + a, & x \leq 0\\ ax + b, & 0 < x < 4\\ \sqrt{x + 21}, & 4 \leq x \end{array} \right .
Continuity
Determine the values of aa and bb such that f(x)f(x) is continuous, where
f(x)={ax+b,if x≤12b+1,if 1<x≤24ax−13b,if x>2f(x)= \begin{cases} ax+b & , \text{if } x \leq 1\\ 2b +1& , \text{if } 1 <x\le2 \\ 4ax-13b&, \text{if } x>2 \end{cases}
When is the following function continuous?
e3x+1sin⁡(πx)\displaystyle \frac{e^{3x}+1}{\sin(\pi x)}