High School
SAT
ACT
University
MCAT
LSAT
DAT

One-sided limits: Evaluating limits by factoring

Related Topics

Wize University Calculus 1 Textbook > Limits

One-Sided Limits

3 Activities

Wize University Calculus 1 Textbook > Limits

Computing Limits by Factoring

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

Evaluate lim⁡x→2−x2−3x+2∣x−2∣\lim_{x\rightarrow2^-}\frac{x^2-3x+2}{\left|x-2\right|}


More One-Sided Limits Questions:
One-sided Limits
Evaluate the limit lim⁡x→1x2−1∣x−1∣\displaystyle\lim_{x\to1}\frac{x^2-1}{\left|x-1\right|}
Practice: One-Sided Limits
Find:
a)lim⁡x→2+∣x−2∣x−2\displaystyle\lim_{x\rightarrow 2^+}\frac{|x-2|}{x-2}
b) lim⁡x→2−∣x−2∣x−2\displaystyle\lim_{x\rightarrow 2^-}\frac{|x-2|}{x-2}
Practice: One-Sided Limits
Find:
a)lim⁡x→2+∣x−2∣x−2\displaystyle\lim_{x\rightarrow 2^+}\frac{|x-2|}{x-2}
b) lim⁡x→2−∣x−2∣x−2\displaystyle\lim_{x\rightarrow 2^-}\frac{|x-2|}{x-2}
One-sided Limits
Evaluate the limit lim⁡x→1x2−1∣x−1∣\displaystyle\lim_{x\to1}\frac{x^2-1}{\left|x-1\right|}
Limit with Absolute Value
Evaluate lim⁡x→2 x−2∣5x−10∣\displaystyle\lim_{x\to2}\ \frac{x-2}{|5x-10|}, if it exists.
Limit with Absolute Value
Evaluate lim⁡x→2 x−2∣5x−10∣\displaystyle\lim_{x\to2}\ \frac{x-2}{|5x-10|}, if it exists.
Limit with Absolute Value
Evaluate lim⁡x→2 x−2∣5x−10∣\displaystyle\lim_{x\to2}\ \frac{x-2}{|5x-10|}, if it exists.
One-sided Limits: Direct Substitution
Practice: Direct Substitution
Given the following function, evaluate the limits as x approaches -1, 2, and 3.
f(x)={−3x+5if  x<−1−10if  x=−19−x2if  −1<x≤21−∣2x−3∣if  x>2f\left(x\right)= \begin{cases} -3x+5 & \text{if } \ x<-1 \\ -10 & \text{if }\ x=-1\\ 9-x^2 & \text{if }\ -1< x\le2\\ 1-|2x-3| & \text{if } \ x>2 \end{cases}
Practice: One-Sided Limits
Find:
a)lim⁡x→2+∣x−2∣x−2\displaystyle\lim_{x\rightarrow 2^+}\frac{|x-2|}{x-2}
b) lim⁡x→2−∣x−2∣x−2\displaystyle\lim_{x\rightarrow 2^-}\frac{|x-2|}{x-2}
One-sided Limits: Direct Substitution
Practice: Direct Substitution
Given the following function, what should the limits be as x approaches -1, 2, and 3?
f(x)={−3x+5if  x<−1−10if  x=−19−x2if  −1<x≤21−∣2x−3∣if  x>2f\left(x\right)= \begin{cases} -3x+5 & \text{if } \ x<-1 \\ -10 & \text{if }\ x=-1\\ 9-x^2 & \text{if }\ -1< x\le2\\ 1-|2x-3| & \text{if } \ x>2 \end{cases}
Practice: Direct Substitution
Practice: Direct Substitution
Given the following function, evaluate the limits below.
f(x)={−3x+5if  x<−1−10if  x=−19−x2if  −1<x≤21−∣2x−3∣if  x>2f\left(x\right)= \begin{cases} -3x+5 & \text{if } \ x<-1 \\ -10 & \text{if }\ x=-1\\ 9-x^2 & \text{if }\ -1< x\le2\\ 1-|2x-3| & \text{if } \ x>2 \end{cases}
Indeterminate Forms: One-sided limits
Evaluate:
lim⁡x→1−∣x−1∣x−1\lim_{x \rightarrow 1^-} \frac{|x-1|}{x-1}
One-Sided Limits
Evaluate
lim⁡x→3−∣x−3∣x2−3\lim\limits_{x\to 3^-} \frac{|x-3|}{x^2-3}
Practice: Direct Substitution
Practice: Direct Substitution
Given the following function, evaluate the limits below.
f(x)={−3x+5if  x<−1−10if  x=−19−x2if  −1<x≤21−∣2x−3∣if  x>2f\left(x\right)= \begin{cases} -3x+5 & \text{if } \ x<-1 \\ -10 & \text{if }\ x=-1\\ 9-x^2 & \text{if }\ -1< x\le2\\ 1-|2x-3| & \text{if } \ x>2 \end{cases}
Practice: Limit of a Function
Example: Limit of a Function
The following is the graph of the function f(x)f\left(x\right)
Determine the following limits:
Consider the function f(x)={12x+aifx<28x2+a−1ifx≥2f\left(x\right)=\begin{cases}\dfrac{1}{2}x+a &\text{if} &x<2\\ \dfrac{8}{x^2+a-1}&\text{if} &x\geq2\\ \end{cases} . What are the values of a for which the limit lim⁡x→2f(x)\displaystyle\lim_{x\to2}f\left(x\right) exists?
Limits: Practice
Find the following limit:
lim⁡x→2−5(x+1)x−2\displaystyle\lim_{x\rightarrow 2}\frac{-5(x+1)}{x-2}
Limits: Practice
Find the following limit:
lim⁡x→2−5(x+1)x−2\displaystyle\lim_{x\rightarrow 2}\frac{-5(x+1)}{x-2}
Evaluate the following limits and the value of the function, where the graphs of f(x) and g(x) are given in the figure. If the limit does not exist, input "DNE".
a) lim⁡x→1f(x)=\displaystyle\lim_{x\rightarrow1}f(x)= b)lim⁡x→2−g(x)=\displaystyle\lim_{x\rightarrow2^-}g(x)=
c)lim⁡x→1+f(x)=\displaystyle\lim_{x\rightarrow1^+}f(x)= d)lim⁡x→2+g(x)=\displaystyle\lim_{x\rightarrow2^+}g(x)=
Let us call all the functions in the figure below as y(x)y\left(x\right). Evaluate the following limits, if they exist. If the limit does not exist, write "DNE".
A) lim⁡x→−1 y(x)\displaystyle\lim_{x\rightarrow-1}\ y(x)
B) lim⁡x→2 y(x)\displaystyle\lim_{x\rightarrow2}\ y(x)
Practice: One-Sided Limits
Find:
a)lim⁡x→2+∣x−2∣x−2\displaystyle\lim_{x\rightarrow 2^+}\frac{|x-2|}{x-2}
b) lim⁡x→2−∣x−2∣x−2\displaystyle\lim_{x\rightarrow 2^-}\frac{|x-2|}{x-2}
One-sided Limits
Evaluate the limit lim⁡x→1x2−1∣x−1∣\displaystyle\lim_{x\to1}\frac{x^2-1}{\left|x-1\right|}
One-sided Limits: Direct Substitution
Practice: Direct Substitution
Given the following function, evaluate the limits below.
f(x)={−3x+5if  x<−1−10if  x=−19−x2if  −1<x≤21−∣2x−3∣if  x>2f\left(x\right)= \begin{cases} -3x+5 & \text{if } \ x<-1 \\ -10 & \text{if }\ x=-1\\ 9-x^2 & \text{if }\ -1< x\le2\\ 1-|2x-3| & \text{if } \ x>2 \end{cases}
One-sided limits: Evaluating limits by factoring
Practice: Evaluating limits
Evaluate lim⁡x→2−x2−3x+2∣x−2∣\lim_{x\rightarrow2^-}\frac{x^2-3x+2}{\left|x-2\right|}
Limit with Absolute Value
Evaluate lim⁡x→2 x−2∣5x−10∣\displaystyle\lim_{x\to2}\ \frac{x-2}{|5x-10|}, if it exists.
One-Sided Limits
If lim⁡x→a−f(x)=2L+3\lim_{x \rightarrow a^-}f(x) = 2L + 3 and lim⁡x→a+f(x)=3L−1\lim_{x \rightarrow a^+}f(x) = 3L - 1 and lim⁡x→af(x)\lim_{x \rightarrow a}f(x) exists with LL a real number, what is the value of LL ?
Practice: One-Sided Limits
Find:
a)lim⁡x→2+∣x−2∣x−2\displaystyle\lim_{x\rightarrow 2^+}\frac{|x-2|}{x-2}
b) lim⁡x→2−∣x−2∣x−2\displaystyle\lim_{x\rightarrow 2^-}\frac{|x-2|}{x-2}
More Computing Limits by Factoring Questions:
Limits: Factoring Terms
lim⁡x→1x5−1x−1\displaystyle\lim_{x\rightarrow1}\dfrac{x^5-1}{x-1}
Limits: Factoring Terms
lim⁡x→1x5−1x−1\displaystyle\lim_{x\rightarrow1}\dfrac{x^5-1}{x-1}
Limits: Factoring
Evaluate
lim⁡x→27x2/3−9x1/3−3\lim\limits_{x\to 27} \frac{x^{2/3}-9}{x^{1/3}-3}
Limits: Factoring
Evaluate
lim⁡x→01−cos⁡2xsin⁡22x\lim\limits_{x\to 0} \frac{1-\cos 2x}{\sin^22x}
Limits: Factoring Terms
lim⁡x→1x5−1x−1\displaystyle\lim_{x\rightarrow1}\dfrac{x^5-1}{x-1}
Limit Techniques: Factoring
lim⁡x→3x−3x3−27\displaystyle \lim_{x\rightarrow 3}\frac{x-3}{x^3-27}
Limit Techniques: Factoring
lim⁡x→3x−3x3−27\displaystyle \lim_{x\rightarrow 3}\frac{x-3}{x^3-27}
Limit Techniques: Factoring
lim⁡x→3x−3x3−27\displaystyle \lim_{x\rightarrow 3}\frac{x-3}{x^3-27}
The limit lim⁡x→2x2−5x+6x2−4\displaystyle\lim_{x\to2}\frac{x^2-5x+6}{x^2-4} evaluates to:
Find the following limit: lim⁡x→5x2−25x−5\displaystyle\lim_{x\rightarrow5}\frac{x^2-25}{x-5}
Limits: Factoring
Evaluate lim⁡x→01−cos⁡2xsin⁡22x\displaystyle \lim_{x\rightarrow 0}\frac{1-\cos{2x}}{\sin^2{2x}}.
Limit with Fractional Powers
Evaluate lim⁡x→27x2/3−9x1/3−3\displaystyle \lim_{x\rightarrow 27}\frac{x^{2/3}-9}{x^{1/3}-3}.
Limits: Cubic
Find the limit lim⁡x→1x3−1x−1\displaystyle\lim\limits_{x\to 1}\frac{x^{3}-1}{x-1}
Limits: Factoring Terms
lim⁡x→1x5−1x−1\displaystyle\lim_{x\rightarrow1}\dfrac{x^5-1}{x-1}
Limits: Factoring Terms
lim⁡x→2x3−5x2+11x−10x−2\displaystyle\lim_{x\rightarrow2}\frac{x^3-5x^2+11x-10}{x-2}
lim⁡x→1x3−14x2+7x+6x−1\displaystyle\lim_{x\rightarrow1}\frac{x^3-14x^2+7x+6}{x-1}
Limits: Factoring Terms
lim⁡x→1x9−1x−1\displaystyle\lim_{x\rightarrow1}\frac{x^9-1}{x-1}
Limits: Factoring Terms
lim⁡x→3x4−81x−3\displaystyle\lim_{x\rightarrow3}\frac{x^4-81}{x-3}
lim⁡x→3x4−81x−3\displaystyle\lim_{x\rightarrow3}\frac{x^4-81}{x-3}
lim⁡x→1x5−1x−1\displaystyle\lim_{x\rightarrow1}\dfrac{x^5-1}{x-1}
Practice: Factorable Indeterminate Forms
Cancelling Common Factors: Evaluate lim⁡x→2x2+x−6x−2\lim_{x\to 2}\limits\dfrac{x^2 + x - 6}{x - 2}
Limits: Factoring
Evaluate
lim⁡x→27x2/3−9x1/3−3\lim\limits_{x\to 27} \frac{x^{2/3}-9}{x^{1/3}-3}
Limits: Factoring
Evaluate
lim⁡x→01−cos⁡2xsin⁡22x\lim\limits_{x\to 0} \frac{1-\cos 2x}{\sin^22x}
Limits: Factoring
Evaluate lim⁡x→01−cos⁡2xsin⁡22x\displaystyle \lim_{x\rightarrow 0}\frac{1-\cos{2x}}{\sin^2{2x}}.
Limit with Fractional Powers
Evaluate lim⁡x→27x2/3−9x1/3−3\displaystyle \lim_{x\rightarrow 27}\frac{x^{2/3}-9}{x^{1/3}-3}.
Limits: Cubic
Find the limit lim⁡x→1x3−1x−1\displaystyle\lim\limits_{x\to 1}\frac{x^{3}-1}{x-1}
Find the following limit: lim⁡x→5x2−25x−5\displaystyle\lim_{x\rightarrow5}\frac{x^2-25}{x-5}
The limit lim⁡x→2x2−5x+6x2−4\displaystyle\lim_{x\to2}\frac{x^2-5x+6}{x^2-4} evaluates to:
The value of the limit lim⁡x→1x−1x2−3x+2\lim_{x\to1}\frac{\sqrt{x}-1}{x^2-3x+2} is
Limits: Factoring (Common Denominator)
Solve the limit lim⁡x→5x−x+10x−225−x2\displaystyle \lim\limits_{x\to 5}\frac{x-\frac{x+10}{x-2}}{25-x^2}.
Limits: Factoring
What is the limit lim⁡x→0sin⁡2xcos⁡x1−cos⁡x\lim\limits_{x\to0}\frac{\sin^2 x\cos x}{1-\cos x}?
The limit lim⁡x→2x2−5x+6x2−4\lim\limits_{x\rightarrow 2}\frac{x^2-5x+6}{x^2-4} evaluates to
Limits: Factoring terms
Evaluate the following limit
lim⁡x→4x(x−2)x−6x+8\displaystyle \lim_{x\rightarrow 4}\frac{x(\sqrt{x}-2)}{x-6\sqrt{x}+8}
Evaluate lim⁡x→2−x2−3x+2∣x−2∣\lim_{x\rightarrow2^-}\frac{x^2-3x+2}{\left|x-2\right|}.
🦊 TRICKY! Determine the value of lim⁡x→−1x3+1x+1\displaystyle \lim_{x\rightarrow -1}\frac{x^3+1}{x+1}.
🦊 TRICKY! Determine the value of lim⁡x→2x3−8x−2\displaystyle \lim_{x \to 2} \frac{x^3-8}{x-2}.
Evaluate lim⁡x→∞4x2+1−2x\displaystyle \lim_{x\rightarrow \infty}\sqrt{4x^2+1}-2x.
Evaluate the limit, lim⁡x→3x2−9x−3\displaystyle\lim_{x\rightarrow3}\frac{x^2-9}{x-3}.
Evaluate the limit, lim⁡x→2x+7−3x−2\displaystyle \lim_{x\rightarrow 2}\frac{\sqrt{x+7}-3}{x-2}.
Evaluate the limit, lim⁡x→0x2+x−2−x\displaystyle \lim_{x\rightarrow0}\frac{x}{\sqrt{2+x}-\sqrt{2-x}}.
Find the following limit: lim⁡x→3x2−9x−3\displaystyle \lim_{x\rightarrow3}\frac{x^2-9}{\sqrt{x}-\sqrt{3}}.
Limits: Factoring
Evaluate the limit lim⁡x→8x3−2x2−64\displaystyle \lim\limits_{x\to 8}\frac{\sqrt[3]{x}-2}{x^2-64}.
One-sided limits: Evaluating limits by factoring
Practice: Evaluating limits
Evaluate lim⁡x→2−x2−3x+2∣x−2∣\lim_{x\rightarrow2^-}\frac{x^2-3x+2}{\left|x-2\right|}
Limits by Factoring Terms
Practice: 0/0 Limit
lim⁡h→0 4−(5h+2)2h=\displaystyle\lim_{h\to0}\ \frac{4-\left(5h+2\right)^2}{h}=
Evaluating Limits: Factoring terms
Evaluate the following limit, or state that it does not exist and why.
lim⁡x→1x3−2x2−5x+6x2−1\lim_{x \rightarrow 1} \frac{x^3 - 2x^2 - 5x + 6}{x^2 - 1}
Limit Techniques: Factoring
lim⁡x→3x−3x3−27\displaystyle \lim_{x\rightarrow 3}\frac{x-3}{x^3-27}
lim⁡x→1x5−1x−1\displaystyle\lim_{x\rightarrow1}\dfrac{x^5-1}{x-1}