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Wize AP Calculus (BC) Textbook > Limits & Continuity

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.