High School
SAT
ACT
University
MCAT
LSAT
DAT
Wize University Calculus 1 Textbook > Limits

One-Sided Limits

Limit LawsPrevious Section
Infinite LimitsNext Section
Popular Courses
Find My Course
0:00 / 0:00

One-Sided Limits

Sometimes we are only concerned with approaching an xx value from one side. This idea leads us to left and right hand limits.

Right-Hand Limit

If f(x)f(x) approaches LL as xx approaches aa and x>ax > a,then we say that LL is the right-hand limit of ffand write
lim⁡x→a+f(x)=L.\displaystyle\boxed{\lim_{x\rightarrow a^+}f(x)=L.}

Left-Hand Limit

If f(x)f(x) approachesLL as xx approaches aa and x<ax < a ,then we say that LL is the left-hand limit of ffand write
lim⁡x→a−f(x)=L.\displaystyle\boxed{\lim_{x\rightarrow a^-}f(x)=L.}

When the Left and Right Limits are Equal

The limit of f(x)f(x) as xx approaches aa exists if both left and right limits exist and are equal. In that case, we have
lim⁡x→a−f(x)=lim⁡x→a+f(x)=lim⁡x→af(x)=L\displaystyle\boxed{\lim_{x\rightarrow a^-}f(x)=\displaystyle\lim_{x\rightarrow a^+}f(x)=\displaystyle\lim_{x\rightarrow a}f(x)=L}


Watch Out!
If the left hand limit does not equal the right hand limit then the two sided limit does not exisit!

0:00 / 0:00

Example: One-Sided Limits


The function ff is defined by
f(x)={−3x+2Ifx≤12x3Ifx>1f(x)=\left\{\begin{array}{l}-3x+2\quad\text{If}\quad x\leq1\\ 2x^3\quad\quad\quad \text{If}\quad x>1\end{array}\right.
Compute the left and right limits of f.f. Does lim⁡x→1f(x)\displaystyle\lim_{x\rightarrow 1}f(x) exist ?




lim⁡x→1−f(x)=(−3)(1)+2=−1lim⁡x→1+f(x)=2(1)3=2},−1≠ 2 ∴lim⁡x→1f(x)DNE\left.\begin{array}{l}\displaystyle\lim_{x\rightarrow1^-}f(x)=(-3)(1)+2=-1\\\displaystyle\lim_{x\rightarrow1^+}f(x)=2(1)^3=2\end{array}\right\},-1\neq\ 2\ \therefore\displaystyle\lim_{x\rightarrow1}f(x) DNE
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}

c) lim⁡x→2∣x−2∣x−2\displaystyle\lim_{x\rightarrow 2}\frac{|x-2|}{x-2}

If the limit does not exist, type DNE
Extra Practice
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: 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|}
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}