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Wize AP Calculus (BC) Textbook > Limits & Continuity

One-Sided Limits

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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!

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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