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Wize University Calculus 1 Textbook > Limits

Basics

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Limit Basics

When we talk about limits we look for what is happening close to a function at a certain value of xx. The question is: What happens to a function's output (the yy value) as we approach a certain input (the xx value)?

Definition of a Limit

Let f(x)f\left(x\right) be a function and let aa and LL be real numbers.

If f(x)f\left(x\right) approaches LL as xx approaches aa then we say f(x)f\left(x\right) has limit LL and denote it as

lim⁡x→af(x)=L\displaystyle\boxed{\lim_{x\rightarrow a}f(x)=L}

Note: We read this as "the limit as xx approaches aa of f(x)f(x) equals LL".


Watch Out!
  • aa does not have to be in the domain of f(x)f(x)
  • xx does not necessarily ever need to be equal to aa (just approach it)

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Example: Limits

Find the value of lim⁡x→0f(x)\displaystyle\lim_{x\rightarrow0}f(x) where
f(x)={0,if x≤21,if x>2f(x)= \begin{cases} 0 & , \text{if } x \leq 2\\ 1 & , \text{if } x > 2 \end{cases}

Notice how as xx "approaches" 00, we are located in the first row of the piecewise function (x≤2)\left(x\le2\right).
So, lim⁡x→0f(x)=0\displaystyle\lim_{x\rightarrow0}f(x)=0.
Extra Practice
Infinite Limits
Evaluate lim⁡x→∞arctan⁡(x−x3)\displaystyle \lim_{x\to\infty}\arctan\left(x-x^3\right)
Limits: Basics
lim⁡x→0∣2x−1∣−∣2x+1∣x\displaystyle\lim_{x\rightarrow0}\dfrac{\lvert 2x-1\rvert-\lvert 2x+1\rvert}{x}
Infinite Limits
Evaluate lim⁡x→∞arctan⁡(x−x3)\displaystyle \lim_{x\to\infty}\arctan\left(x-x^3\right)
Limit with Absolute Value
lim⁡x→0 ∣3−2x∣−∣2x−3∣x=\displaystyle\lim_{x\to0}\ \frac{\left|3-2x\right|-\left|2x-3\right|}{x}=
Limits: Basics
lim⁡x→0∣2x−1∣−∣2x+1∣x\displaystyle\lim_{x\rightarrow0}\dfrac{\lvert 2x-1\rvert-\lvert 2x+1\rvert}{x}
Limits: Basics
Determine lim⁡x→π2ln⁡(sin⁡x)\displaystyle \lim_{x\to\frac{\pi}{2}}\ln\left(\sin x\right).
Inverse Trig & Exponential Limits
Find the value of lim⁡x→0+arctan⁡(e1x)\displaystyle\lim_{x\rightarrow0^+}\arctan\left(e^{\frac{1}{x}}\right).
Basic limit techniques - Absolute value
Evaluate lim⁡x→4−∣x−4∣x2−4.\lim\limits_{x\rightarrow4^-}\frac{\vert x-4\vert}{x^2-4}.
Basic limit techniques - Absolute value
Evaluate lim⁡x→4−∣x−4∣x2−4.\lim\limits_{x\rightarrow4^-}\frac{\vert x-4\vert}{x^2-4}.
Inverse Trig & Exponential Limits
Find the value of lim⁡x→0+arctan⁡(e1x)\displaystyle\lim_{x\rightarrow0^+}\arctan\left(e^{\frac{1}{x}}\right).
Inverse Trig & Exponential Limits
Find the value of lim⁡x→0+arctan⁡(e1x)\displaystyle\lim_{x\rightarrow0^+}\arctan\left(e^{\frac{1}{x}}\right).
Limits: Absolute Values
Q.\textbf{Q.} Evaluate lim⁡x→0∣2x−1∣−∣2x+1∣x\displaystyle \lim_{x\rightarrow 0}\frac{\lvert 2x-1\rvert-\lvert 2x+1\rvert}{x}.
Limit with Absolute Value
lim⁡x→0 ∣3−2x∣−∣2x−3∣x=\displaystyle\lim_{x\to0}\ \frac{\left|3-2x\right|-\left|2x-3\right|}{x}=
Infinite Limits
Evaluate lim⁡x→∞arctan⁡(x−x3)\displaystyle \lim_{x\to\infty}\arctan\left(x-x^3\right)
Inverse Trig & Exponential Limits
Find the value of lim⁡x→0+arctan⁡(e1x)\displaystyle\lim_{x\rightarrow0^+}\arctan\left(e^{\frac{1}{x}}\right).
Trigonometric Functions
Evaluate lim⁡x→0sin⁡(1∣x∣)\lim_{x\rightarrow0}\sin\left(\frac{1}{\left|x\right|}\right).
Limits: Absolute Values
Q.\textbf{Q.} Evaluate lim⁡x→0∣2x−1∣−∣2x+1∣x\displaystyle \lim_{x\rightarrow 0}\frac{\lvert 2x-1\rvert-\lvert 2x+1\rvert}{x}.
Basic limit techniques - Absolute value
Evaluate lim⁡x→4−∣x−4∣x2−4.\lim\limits_{x\rightarrow4^-}\frac{\vert x-4\vert}{x^2-4}.
Limits: Basics
lim⁡x→0∣2x−1∣−∣2x+1∣x\displaystyle\lim_{x\rightarrow0}\dfrac{\lvert 2x-1\rvert-\lvert 2x+1\rvert}{x}
Limits: Basics
Determine lim⁡x→π2ln⁡(sin⁡x)\displaystyle \lim_{x\to\frac{\pi}{2}}\ln\left(\sin x\right).
lim⁡x→0∣2x−1∣−∣2x+1∣x\displaystyle\lim_{x\rightarrow0}\dfrac{\lvert 2x-1\rvert-\lvert 2x+1\rvert}{x}