Wize AP Calculus (BC) Textbook > Limits & Continuity

Infinite Limits

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Infinite Limits
Infinite Limits describe the behavior of functions whose values become arbitrarily large either positively or negatively. The answer to these limits is  ±∞\pm\infty±∞. 


(Positive) Infinite Limits
Let  f(x)f\left(x\right)f(x) be a function defined on both sides of aaa except possibly at aaa itself. If f(x)f(x)f(x) grows arbitrarily large as  xxx approaches aaa then 
lim⁡x→af(x)=∞ \displaystyle \boxed{\lim_{x \to a} f(x)=\infty}x→alim​f(x)=∞​

(Negative) Infinite Limits
Let f(x)f(x)f(x) be a function defined on both sides of aaa except possibly at aaa itself. If f(x)f(x)f(x) grows arbitrary small as xxxapproaches aaa then
lim⁡x→af(x)=−∞ \displaystyle \boxed{\lim_{x \to a} f(x)=-\infty}x→alim​f(x)=−∞​


Wize Tip
When trying to evaluate infinite limits at aaa ,you will obtain a 0 in the denominator. Your job is to figure out if the function is growing positively or negatives close to aaa.


Vertical Asymptotes
A function  f(x)f\left(x\right)f(x) has a vertical asymptote with equation  x=ax=ax=a  if at least one of these statements is true:

lim⁡x→a−f(x)=±∞    or    lim⁡x→a+f(x)=±∞\displaystyle \boxed{\lim_{x \to a^-} f(x)= \pm \infty \;\; \text{or}\;\;  \lim_{x \to a^+} f(x)= \pm \infty }x→a−lim​f(x)=±∞orx→a+lim​f(x)=±∞​

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Example: Infinite Limits
For the following graph, find:

i) lim⁡x→5+f(x)\displaystyle \lim_{x \to 5^+}f(x)x→5+lim​f(x)

ii)lim⁡x→5−f(x)\displaystyle \lim_{x \to 5^-}f(x)x→5−lim​f(x)
Note: The graph above has a vertical asymptote at x=5x=5x=5.

i) lim⁡x→5+f(x)=∞\displaystyle \lim_{x \to 5^+}f(x)=\infinx→5+lim​f(x)=∞

ii) lim⁡x→5−f(x)=−∞\displaystyle \lim_{x \to 5^-}f(x)=-\infinx→5−lim​f(x)=−∞

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Example: Infinite Limits
Find the following limit

lim⁡x→73+7x−63x−7\displaystyle \lim_{x \to \frac{7}{3}^+} \frac{7x-6}{3x-7}x→37​+lim​3x−77x−6​ 

When we try to evaluate the limit, the numerator is positive while the denominator becomes 0. Evaluating with numbers to the right of 7/3 will give positive values in the denominator. 

=493−60+=+0+=+∞=\displaystyle\frac{\frac{49}{3}-6}{0^+}=\frac{+}{0^+}=+\infty=0+349​−6​=0++​=+∞