Infinite Limits and Asymptote Practice

$\displaystyle\lim_{x\to4^-}\frac1{x-4}$

$\displaystyle\lim_{x\to4^+}\frac1{x-4}$

$\displaystyle\lim_{x\to0^-}\frac1{x^2}$

$\displaystyle\lim_{x\to0^+}\frac1{x^3}$

$\displaystyle\lim_{x\to-2^+}\frac{x-1}{x+2}$

$\displaystyle\lim_{x\to1^-}\frac{x+3}{(x-1)^2}$

$\displaystyle\lim_{x\to3^-}\frac{x+2}{(x-3)(x+1)}$

$\displaystyle\lim_{x\to3^+}\frac{x+2}{(x-3)(x+1)}$

$\displaystyle\lim_{x\to-1^-}\frac{2-x}{(x+1)^3}$

$\displaystyle\lim_{x\to-1^+}\frac{2-x}{(x+1)^3}$

Find and classify discontinuities of $\dfrac{x-2}{(x-2)(x+1)}$.

Find and classify discontinuities of $\dfrac{x^2-9}{(x-3)(x+2)}$.

Find all holes and vertical asymptotes of $\dfrac{(x+1)(x-4)}{(x+1)(x-2)^2}$.

Determine one-sided behavior at every vertical asymptote of $\dfrac{x+3}{(x-1)(x+2)}$.

Determine one-sided behavior at every vertical asymptote of $\dfrac{x-5}{(x+1)^2}$.

Explain why a cancelled denominator factor creates a hole rather than a vertical asymptote.

$\displaystyle\lim_{x\to\infty}\frac{4x+1}{x^2+7}$

$\displaystyle\lim_{x\to-\infty}\frac{2x^2-3}{5x^2+x}$

$\displaystyle\lim_{x\to\infty}\frac{7x^3+x}{2x^3-5}$

$\displaystyle\lim_{x\to-\infty}\frac{3x^4-x}{x^4+9}$

$\displaystyle\lim_{x\to\infty}\frac{x^2+1}{x^3+2}$

$\displaystyle\lim_{x\to-\infty}\frac{5x^3+1}{x^2-4}$

$\displaystyle\lim_{x\to\infty}\frac{-2x^5+x}{x^5+3}$

Find all horizontal asymptotes of $\dfrac{3x^2+1}{x^2-5x}$.

Find the slant asymptote of $\dfrac{x^2+3}{x-2}$.

Use division to describe the end behavior of $\dfrac{2x^3+x}{x^2-1}$.

$\displaystyle\lim_{x\to\infty}\frac{\sqrt{x^2+4}}{x}$

$\displaystyle\lim_{x\to-\infty}\frac{\sqrt{x^2+4}}{x}$

$\displaystyle\lim_{x\to\infty}\frac{\sqrt{9x^2+1}}{x}$

$\displaystyle\lim_{x\to-\infty}\frac{\sqrt{9x^2+1}}{x}$

$\displaystyle\lim_{x\to\infty}\left(\sqrt{x^2+8x}-x\right)$

$\displaystyle\lim_{x\to\infty}\left(\sqrt{4x^2+x}-2x\right)$

$\displaystyle\lim_{x\to-\infty}\left(\sqrt{x^2+6x}+x\right)$

Explain the exact point where $|x|$ must be used in a radical-at-infinity problem.

Analyze all discontinuities and asymptotes of $\dfrac{x^2-1}{x^2-3x+2}$.

Analyze all discontinuities and asymptotes of $\dfrac{x^2-4}{x^2-x-2}$.

Find the exact one-sided limits at every vertical asymptote of $\dfrac{x}{x^2-9}$.

Find the end behavior of $\dfrac{2x^3-x+1}{x^2+4}$, including any polynomial asymptote.

Construct a rational function with a hole at $x=1$, a vertical asymptote at $x=-2$, and horizontal asymptote $y=3$.

Explain why a graph may cross a horizontal asymptote but cannot take a finite value on a vertical asymptote where its formula is undefined.