4.5: Limits at Infinity and Asymptotes

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Evaluate $\displaystyle \lim_{x \to \infty} (5x^2 - 3x + 1)$.
Evaluate $\displaystyle \lim_{x \to -\infty} (5x^2 - 3x + 1)$.
Evaluate $\displaystyle \lim_{x \to \infty} (2 + x - 3x^7)$.
Evaluate $\displaystyle \lim_{x \to -\infty} (2 + x - 3x^7)$.
Evaluate $\displaystyle \lim_{x \to \infty} \dfrac{1}{3x + 6}$.
Evaluate $\displaystyle \lim_{x \to -\infty} \dfrac{1}{3x + 6}$.
Evaluate $\displaystyle \lim_{x \to \infty} \dfrac{4x + 1}{2x - 5}$.
Evaluate $\displaystyle \lim_{x \to -\infty} \dfrac{4x + 1}{2x - 5}$.
Evaluate $\displaystyle \lim_{x \to \infty} \dfrac{x^2 - 2x + 5}{3x + 7}$.
Evaluate $\displaystyle \lim_{x \to -\infty} \dfrac{x^2 - 2x + 5}{3x + 7}$.
Evaluate $\displaystyle \lim_{x \to \infty} \dfrac{2 - 4x^3 + 7x^5}{ 1 + 3x^5}$.
Evaluate $\displaystyle \lim_{x \to -\infty} \dfrac{2x^2 - 7x + 1}{1 - x^3}$.
Evaluate $\displaystyle \lim_{x \to -\infty} \dfrac{5x^4 - 3x^2 + x}{10x^2 - 4}$.
Evaluate $\displaystyle \lim_{x \to \infty} \dfrac{2 - 4x^3 + 7x^5}{ 1 + 3x^5}$.
Evaluate $\displaystyle \lim_{x \to \infty} \sqrt{4x^2 - 7x}$.
Evaluate $\displaystyle \lim_{x \to -\infty} \sqrt{4x^2 - 7x}$.
Evaluate $\displaystyle \lim_{x \to \infty} \dfrac{2x + 3}{\sqrt{x^2 + 5x}}$.
Evaluate $\displaystyle \lim_{x \to -\infty} \dfrac{2x + 3}{\sqrt{x^2 + 5x}}$.
Evaluate $\displaystyle \lim_{x \to \infty} (\sqrt{x^2 + 2} - x)$.
Evaluate $\displaystyle \lim_{x \to -\infty} (\sqrt{x^2 + 2} - x)$.
Evaluate $\displaystyle \lim_{x \to \infty} \sin x$.
Evaluate $\displaystyle \lim_{x \to \infty} \dfrac{\sin x}{x}$.
Find all horizontal and vertical asymptotes of $f(x) = \dfrac{2x}{x - 3}$.
Find all horizontal and vertical asymptotes of $f(x) = \dfrac{7x - 2}{x^2 + 5x + 4}$.
Find all horizontal and vertical asymptotes of $f(x) = \dfrac{x + 2}{x^2 - 4}$.
Find all horizontal and vertical asymptotes of $f(x) = \dfrac{\sqrt{9x^2 + 1}}{3x - 5}$.
Given function $f(x)$ graphed below, construct sign charts for $f$, $f'$, and $f''$.
$A graph of the function (x+1)/(x+2).$
Given the function $f(x)$ graphed below, construct sign charts for $f$, $f'$, and $f''$.
$The graph of the function (x-1)^3/((x-1)^4 - 1).$
Given the function $f(x)$ graphed below, construct sign charts for $f$, $f'$, and $f''$.
$A piecewise function equal to -x(x+2) for x≤0 and equal to -x(x-4) for x>0.$
Given the function $f(x)$ graphed below, construct sign charts for $f$, $f'$, and $f''$.
$The graph of (x-1)^(1/3) + 1.$
Sketch the graph of a function $f(x)$ that agrees with the properties and sign charts below.
- $f(x)$ has a vertical asymptote at $x = 2$
- $\displaystyle \lim_{x \to -\infty} f(x) = \lim_{x \to \infty} f(x) = -1$
$Sign charts showing f(x) is negative for x<1, positive for 1<x<2, positive for 2<x<3, and negative for x>3; f'(x) is positive for x<2 and negative for x>2; and f''(x) is positive for x<2 and positive for x>2. In all sign charts, the tick marks for 2 are dashed.$

Sketch the graph of a function $f(x)$ that agrees with the properties and sign charts below.
- $f(x)$ has vertical asymptotes at $x = -3$ and $x = 1$
- $\displaystyle \lim_{x \to -\infty} f(x) = \lim_{x \to \infty} f(x) = 0$
$Sign charts showing f(x) is positive for x<-3, negative for -3<x<-2, negative for -1<x<1, negative for 1<x<2, and positive for x>2; f'9x) is positive for x<-3, positive for -3<x<-1, negative for -1<x<1, positive for 1<x<4, and negative for x>4; and f''(x) is positive for x<-3, negative for -3<x<1, negative for 1<x<6, and positive for x>6. In all sign charts, the tick marks for -3 and 1 and dashed.$

Sketch the graph of a function $f(x)$ that agrees with the properties and sign charts below.
- $f(x)$ has no vertical asymptotes
- $\displaystyle \lim_{x \to -\infty} f(x) = -\infty$ and $\displaystyle \lim_{x \to \infty} f(x) = \infty$
$Sign charts showing f(x) is negative for x<0 and positive for x>0; f'(x) is positive for x<2, negative for 2<x<3, and positive for x>3; and f''(x) is negative for x<3 and negative for x>3. The tick mark for 3 on the sign charts for f' and f'' is represented by a dotted line.$
Sketch the graph of a function $f(x)$ that agrees with the properties and sign charts below.
- $f(x)$ has no vertical asymptotes
- $\displaystyle \lim_{x \to -\infty} f(x) = 3$ and $\displaystyle \lim_{x \to \infty} f(x) = -1$
$Sign charts showing f(x) is positive for x<-1/2 and negative for x>-1/2; f'(x) is negative for x<-1 and negative for x>-1; f''(x) is negative for x<-1 and positive for x>-1. The tick marks for -1 in the sign charts for f' and f'' are dashed.$
Consider the function $f(x) = \dfrac{3x-3}{2-x}$. Find all vertical and horizontal asymptotes of the function and construct sign charts for $f$, $f'$, and $f''$, then sketch the graph of $f$. Be sure to include all $x$-intercepts, relative extrema, inflection points, and asymptotic behavior in your graph. Confirm your result using a graphing calculator.
Consider the function $f(x) = \dfrac{2}{x^2 - 1}$. Find all vertical and horizontal asymptotes of the function and construct sign charts for $f$, $f'$, and $f''$, then sketch the graph of $f$. Be sure to include all $x$-intercepts, relative extrema, inflection points, and asymptotic behavior in your graph. Confirm your result using a graphing calculator.
Consider the function $f(x) = \dfrac{x}{x^2-1}$. Find all vertical and horizontal asymptotes of the function and construct sign charts for $f$, $f'$, and $f''$, then sketch the graph of $f$. Be sure to include all $x$-intercepts, relative extrema, inflection points, and asymptotic behavior in your graph. Confirm your result using a graphing calculator.
Consider the function $f(x) = \dfrac{x^2}{x^2-1}$. Find all vertical and horizontal asymptotes of the function and construct sign charts for $f$, $f'$, and $f''$, then sketch the graph of $f$. Be sure to include all $x$-intercepts, relative extrema, inflection points, and asymptotic behavior in your graph. Confirm your result using a graphing calculator.
Consider the function $f(x) = \dfrac{x^2}{x^2+1}$. Find all vertical and horizontal asymptotes of the function and construct sign charts for $f$, $f'$, and $f''$, then sketch the graph of $f$. Be sure to include all $x$-intercepts, relative extrema, inflection points, and asymptotic behavior in your graph. Confirm your result using a graphing calculator.
Consider the function $f(x) = \dfrac{1}{x^3+3x^2}$. Find all vertical and horizontal asymptotes of the function and construct sign charts for $f$, $f'$, and $f''$, then sketch the graph of $f$. Be sure to include all $x$-intercepts, relative extrema, inflection points, and asymptotic behavior in your graph. Confirm your result using a graphing calculator.
Consider the function $f(x) = x^{1/3}$. Find all vertical and horizontal asymptotes of the function and construct sign charts for $f$, $f'$, and $f''$, then sketch the graph of $f$. Be sure to include all $x$-intercepts, relative extrema, inflection points, and asymptotic behavior in your graph. Confirm your result using a graphing calculator.
Consider the function $f(x) = x^{2/3}$. Find all vertical and horizontal asymptotes of the function and construct sign charts for $f$, $f'$, and $f''$, then sketch the graph of $f$. Be sure to include all $x$-intercepts, relative extrema, inflection points, and asymptotic behavior in your graph. Confirm your result using a graphing calculator.
Consider the function $f(x) = (x-1)^2(x+3)^{2/3}$. Find all vertical and horizontal asymptotes of the function and construct sign charts for $f$, $f'$, and $f''$, then sketch the graph of $f$. Be sure to include all $x$-intercepts, relative extrema, inflection points, and asymptotic behavior in your graph. Confirm your result using a graphing calculator.
Consider the function $f(x) = \dfrac{2x+2}{\sqrt{x^2+1}}$. Find all vertical and horizontal asymptotes of the function and construct sign charts for $f$, $f'$, and $f''$, then sketch the graph of $f$. Be sure to include all $x$-intercepts, relative extrema, inflection points, and asymptotic behavior in your graph. Confirm your result using a graphing calculator.

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