Chapter 4 Exercises A. One-sided infinite limits lim x → 4 − 1 x − 4 \displaystyle\lim_{x\to4^-}\frac1{x-4} x → 4 − lim x − 4 1
lim x → 4 + 1 x − 4 \displaystyle\lim_{x\to4^+}\frac1{x-4} x → 4 + lim x − 4 1
lim x → 0 − 1 x 2 \displaystyle\lim_{x\to0^-}\frac1{x^2} x → 0 − lim x 2 1
lim x → 0 + 1 x 3 \displaystyle\lim_{x\to0^+}\frac1{x^3} x → 0 + lim x 3 1
lim x → − 2 + x − 1 x + 2 \displaystyle\lim_{x\to-2^+}\frac{x-1}{x+2} x → − 2 + lim x + 2 x − 1
lim x → 1 − x + 3 ( x − 1 ) 2 \displaystyle\lim_{x\to1^-}\frac{x+3}{(x-1)^2} x → 1 − lim ( x − 1 ) 2 x + 3
lim x → 3 − x + 2 ( x − 3 ) ( x + 1 ) \displaystyle\lim_{x\to3^-}\frac{x+2}{(x-3)(x+1)} x → 3 − lim ( x − 3 ) ( x + 1 ) x + 2
lim x → 3 + x + 2 ( x − 3 ) ( x + 1 ) \displaystyle\lim_{x\to3^+}\frac{x+2}{(x-3)(x+1)} x → 3 + lim ( x − 3 ) ( x + 1 ) x + 2
lim x → − 1 − 2 − x ( x + 1 ) 3 \displaystyle\lim_{x\to-1^-}\frac{2-x}{(x+1)^3} x → − 1 − lim ( x + 1 ) 3 2 − x
lim x → − 1 + 2 − x ( x + 1 ) 3 \displaystyle\lim_{x\to-1^+}\frac{2-x}{(x+1)^3} x → − 1 + lim ( x + 1 ) 3 2 − x
B. Holes and vertical asymptotes Find and classify discontinuities of x − 2 ( x − 2 ) ( x + 1 ) \dfrac{x-2}{(x-2)(x+1)} ( x − 2 ) ( x + 1 ) x − 2
Find and classify discontinuities of x 2 − 9 ( x − 3 ) ( x + 2 ) \dfrac{x^2-9}{(x-3)(x+2)} ( x − 3 ) ( x + 2 ) x 2 − 9
Find all holes and vertical asymptotes of ( x + 1 ) ( x − 4 ) ( x + 1 ) ( x − 2 ) 2 \dfrac{(x+1)(x-4)}{(x+1)(x-2)^2} ( x + 1 ) ( x − 2 ) 2 ( x + 1 ) ( x − 4 )
Determine one-sided behavior at every vertical asymptote of x + 3 ( x − 1 ) ( x + 2 ) \dfrac{x+3}{(x-1)(x+2)} ( x − 1 ) ( x + 2 ) x + 3
Determine one-sided behavior at every vertical asymptote of x − 5 ( x + 1 ) 2 \dfrac{x-5}{(x+1)^2} ( x + 1 ) 2 x − 5
Explain why a cancelled denominator factor creates a hole rather than a vertical asymptote.
C. Rational limits at infinity lim x → ∞ 4 x + 1 x 2 + 7 \displaystyle\lim_{x\to\infty}\frac{4x+1}{x^2+7} x → ∞ lim x 2 + 7 4 x + 1
lim x → − ∞ 2 x 2 − 3 5 x 2 + x \displaystyle\lim_{x\to-\infty}\frac{2x^2-3}{5x^2+x} x → − ∞ lim 5 x 2 + x 2 x 2 − 3
lim x → ∞ 7 x 3 + x 2 x 3 − 5 \displaystyle\lim_{x\to\infty}\frac{7x^3+x}{2x^3-5} x → ∞ lim 2 x 3 − 5 7 x 3 + x
lim x → − ∞ 3 x 4 − x x 4 + 9 \displaystyle\lim_{x\to-\infty}\frac{3x^4-x}{x^4+9} x → − ∞ lim x 4 + 9 3 x 4 − x
lim x → ∞ x 2 + 1 x 3 + 2 \displaystyle\lim_{x\to\infty}\frac{x^2+1}{x^3+2} x → ∞ lim x 3 + 2 x 2 + 1
lim x → − ∞ 5 x 3 + 1 x 2 − 4 \displaystyle\lim_{x\to-\infty}\frac{5x^3+1}{x^2-4} x → − ∞ lim x 2 − 4 5 x 3 + 1
lim x → ∞ − 2 x 5 + x x 5 + 3 \displaystyle\lim_{x\to\infty}\frac{-2x^5+x}{x^5+3} x → ∞ lim x 5 + 3 − 2 x 5 + x
Find all horizontal asymptotes of 3 x 2 + 1 x 2 − 5 x \dfrac{3x^2+1}{x^2-5x} x 2 − 5 x 3 x 2 + 1
Find the slant asymptote of x 2 + 3 x − 2 \dfrac{x^2+3}{x-2} x − 2 x 2 + 3
Use division to describe the end behavior of 2 x 3 + x x 2 − 1 \dfrac{2x^3+x}{x^2-1} x 2 − 1 2 x 3 + x
D. Radicals at infinity lim x → ∞ x 2 + 4 x \displaystyle\lim_{x\to\infty}\frac{\sqrt{x^2+4}}{x} x → ∞ lim x x 2 + 4
lim x → − ∞ x 2 + 4 x \displaystyle\lim_{x\to-\infty}\frac{\sqrt{x^2+4}}{x} x → − ∞ lim x x 2 + 4
lim x → ∞ 9 x 2 + 1 x \displaystyle\lim_{x\to\infty}\frac{\sqrt{9x^2+1}}{x} x → ∞ lim x 9 x 2 + 1
lim x → − ∞ 9 x 2 + 1 x \displaystyle\lim_{x\to-\infty}\frac{\sqrt{9x^2+1}}{x} x → − ∞ lim x 9 x 2 + 1
lim x → ∞ ( x 2 + 8 x − x ) \displaystyle\lim_{x\to\infty}\left(\sqrt{x^2+8x}-x\right) x → ∞ lim ( x 2 + 8 x − x )
lim x → ∞ ( 4 x 2 + x − 2 x ) \displaystyle\lim_{x\to\infty}\left(\sqrt{4x^2+x}-2x\right) x → ∞ lim ( 4 x 2 + x − 2 x )
lim x → − ∞ ( x 2 + 6 x + x ) \displaystyle\lim_{x\to-\infty}\left(\sqrt{x^2+6x}+x\right) x → − ∞ lim ( x 2 + 6 x + x )
Explain the exact point where ∣ x ∣ |x| ∣ x ∣
E. Mixed exam practice Analyze all discontinuities and asymptotes of x 2 − 1 x 2 − 3 x + 2 \dfrac{x^2-1}{x^2-3x+2} x 2 − 3 x + 2 x 2 − 1
Analyze all discontinuities and asymptotes of x 2 − 4 x 2 − x − 2 \dfrac{x^2-4}{x^2-x-2} x 2 − x − 2 x 2 − 4
Find the exact one-sided limits at every vertical asymptote of x x 2 − 9 \dfrac{x}{x^2-9} x 2 − 9 x
Find the end behavior of 2 x 3 − x + 1 x 2 + 4 \dfrac{2x^3-x+1}{x^2+4} x 2 + 4 2 x 3 − x + 1
Construct a rational function with a hole at x = 1 x=1 x = 1 x = − 2 x=-2 x = − 2 y = 3 y=3 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.
Answers begin in the referenced section.
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