?Find all vertical asymptotes, x = a, of the following functions. For each value of a

Use a graph to explain the meaning of \(\lim \limits_{x \rightarrow a^{+}} f(x)=-\infty\)

Use a graph to explain the meaning of \(\lim \limits_{x \rightarrow a} f(x)=\infty\).

What is a vertical asymptote?

Consider the function F(x) = f(x)/g(x) with g(a) = 0. Does F necessarily have a vertical asymptote at x = a? Explain your reasoning.

Suppose \(f(x) \rightarrow 100\) and \(g(x) \rightarrow 0\) with \(g(x)<0 \text { as } x \rightarrow 2\). Determine \(\lim \limits_{x \rightarrow 2} \frac{f(x)}{g(x)}\).

Evaluate \(\lim \limits_{x \rightarrow 3^{-}} \frac{1}{x-3}\) and \(\lim \limits_{x \rightarrow 3^{+}} \frac{1}{x-3^{+}}\).

Compute the values of \(f(x)=\frac{x+1}{(x-1)^{2}}\) in the following table and use them to determine \(\lim \limits_{x \rightarrow 1} f(x)\).

Use the graph of \(f(x)=\frac{x}{\left(x^{2}-2 x-3\right)^{2}}\) to determine \(\lim \limits_{x \rightarrow-1} f(x)\) and \(\lim \limits_{x \rightarrow 3} f(x)\).

The graph of f in the figure has vertical asymptotes at x = 1 and x = 2. Find the following limits, if possible. a. \(\lim \limits_{x \rightarrow 1^{-}} f(x)\) b. \(\lim \limits_{x \rightarrow 1^{+}} f(x)\) c. \(\lim \limits_{x \rightarrow 1} f(x)\) d. \(\lim \limits_{x \rightarrow 2^{-}} f(x)\) e. \(\lim \limits_{x \rightarrow 2^{+}} f(x)\) f. \(\lim \limits_{x \rightarrow 2} f(x)\)

The graph of g in the figure has vertical asymptotes at x = 2 and x = 4. Find the following limits, if possible. a. \(\lim \limits_{x \rightarrow 2^{-}} g(x)\) b. \(\lim \limits_{x \rightarrow 2^{+}} g(x)\) c. \(\lim \limits_{x \rightarrow 2} g(x)\) d. \(\lim \limits_{x \rightarrow 4^{-}} g(x)\) e. \(\lim \limits_{x \rightarrow 4^{+}} g(x)\) f. \(\lim \limits_{x \rightarrow 4} g(x)\)

The graph of h in the figure has vertical asymptotes at x = -2 and x = 3. Find the following limits, if possible. a. \(\lim \limits_{x \rightarrow-2^{-}} h(x)\) b. \(\lim \limits_{x \rightarrow-2^{+}} h(x)\) c. \(\lim \limits_{x \rightarrow-2} h(x)\) d. \(\lim \limits_{x \rightarrow 3^{-}} h(x)\) e. \(\lim \limits_{x \rightarrow 3^{+}} h(x)\) f. \(\lim \limits_{x \rightarrow 3} h(x)\)

The graph of p in the figure has vertical asymptotes at x = -2 and x = 3. Find the following limits, if possible. a. \(\lim \limits_{x \rightarrow-2^{-}} p(x)\) b. \(\lim \limits_{x \rightarrow-2^{+}} p(x)\) c. \(\lim \limits_{x \rightarrow-2} p(x)\) d. \(\lim \limits_{x \rightarrow 3^{-}} p(x)\) e. \(\lim \limits_{x \rightarrow 3^{+}} p(x)\) f. \(\lim \limits_{x \rightarrow 3} p(x)\)

Graph the function \(f(x)=\frac{1}{x^{2}-x}\) using a graphing utility with the window \([-1,2] \times[-10,10]\). Use the graph to determine the following limits. a. \(\lim \limits_{x \rightarrow 0^{-}} f(x)\) b. \(\lim \limits_{x \rightarrow 0^{+}} f(x)\) c. \(\lim \limits_{x \rightarrow 1^{-}} f(x)\) d. \(\lim \limits_{x \rightarrow 1^{+}} f(x)\)

Graph the function \(f(x)=\frac{e^{-x}}{x(x+2)^{2}}\) using a graphing utility. (Experiment with your choice of a graphing window.) Use the graph to determine the following limits. a. \(\lim \limits_{x \rightarrow-2^{+}} f(x)\) b. \(\lim \limits_{x \rightarrow-2} f(x)\) c. \(\lim \limits_{x \rightarrow 0^{-}} f(x)\) d. \(\lim \limits_{x \rightarrow 0^{+}} f(x)\)

Sketch a possible graph of a function f, together with vertical asymptotes, satisfying all of the following conditions. f(1) = 0 f(3) is undefined \(\lim \limits_{x \rightarrow 3} f(x)=1\) \(\lim \limits_{x \rightarrow 0^{+}} f(x)=-\infty\) \(\lim \limits_{x \rightarrow 2} f(x)=\infty\) \(\lim \limits_{x \rightarrow 4^{-}} f(x)=\infty\)

Sketch a possible graph of a function g, together with vertical asymptotes, satisfying all of the following conditions. g(2) = 1 g(5) = -1 \(\lim \limits_{x \rightarrow 4} g(x)=-\infty\) \(\lim \limits_{x \rightarrow 7} g(x)=\infty\) \(\lim \limits_{x \rightarrow 7^{+}} g(x)=-\infty\)

Evaluate the following limits, or state that they do not exist. a. \(\lim \limits_{x \rightarrow 2^{+}} \frac{1}{x-2}\) b. \(\lim \limits_{x \rightarrow 2^{-}} \frac{1}{x-2}\) c. \(\lim \limits_{x \rightarrow 2} \frac{1}{x-2}\)

Evaluate the following limits, or state that they do not exist. a. \(\lim \limits_{x \rightarrow 3^{+}} \frac{2}{(x-3)^{3}}\) b. \(\lim \limits_{x \rightarrow 3^{-}} \frac{2}{(x-3)^{3}}\) c. \(\lim \limits_{x \rightarrow 3} \frac{2}{(x-3)^{3}}\)

Evaluate the following limits, or state that they do not exist. \(\lim \limits_{x \rightarrow 0} \frac{x^{3}-5 x^{2}}{x^{2}}\)

Evaluate the following limits, or state that they do not exist. \(\lim \limits_{t \rightarrow 5} \frac{4 t^{2}-100}{t-5}\)

Evaluate the following limits, or state that they do not exist. \(\lim \limits_{x \rightarrow 1^{+}} \frac{x^{2}-5 x+6}{x-1}\)

Evaluate the following limits, or state that they do not exist. \(\lim \limits_{z \rightarrow 4} \frac{z-5}{\left(z^{2}-10 z+24\right)^{2}}\)

Find all vertical asymptotes, x = a, of the following functions. For each value of a, evaluate \(\lim \limits_{x \rightarrow a^{+}} f(x)\), \(\lim \limits_{x \rightarrow a^{-}} f(x)\), and \(\lim \limits_{x \rightarrow a} f(x)\). \(f(x)=\frac{x^{2}-9 x+14}{x^{2}-5 x+6}\)

Find all vertical asymptotes, x = a, of the following functions. For each value of a, evaluate \(\lim \limits_{x \rightarrow a^{+}} f(x)\), \(\lim \limits_{x \rightarrow a^{-}} f(x)\), and \(\lim \limits_{x \rightarrow a} f(x)\). \(f(x)=\frac{\cos x}{x^{2}+2 x}\)

Find all vertical asymptotes, x = a, of the following functions. For each value of a, evaluate \(\lim \limits_{x \rightarrow a^{+}} f(x)\), \(\lim \limits_{x \rightarrow a^{-}} f(x)\), and \(\lim \limits_{x \rightarrow a} f(x)\). \(f(x)=\frac{x+1}{x^{3}-4 x^{2}+4 x}\)

Find all vertical asymptotes, x = a, of the following functions. For each value of a, evaluate \(\lim \limits_{x \rightarrow a^{+}} f(x)\), \(\lim \limits_{x \rightarrow a^{-}} f(x)\), and \(\lim \limits_{x \rightarrow a} f(x)\). \(f(x)=\frac{x^{3}-10 x^{2}+16 x}{x^{2}-8 x}\)

Evaluate the following limits. \(\lim _{\theta \rightarrow 0^{+}} \csc \theta\)

Evaluate the following limits. \(\lim \limits_{x \rightarrow 0^{-}} \csc x\)

Evaluate the following limits. \(\lim \limits_{x \rightarrow 0^{+}}(-10 \cot x)\)

Evaluate the following limits. \(\lim \limits_{\theta \rightarrow \pi / 2^{+}} \frac{1}{3} \tan \theta\)

Graph the function y = tan x with the window \([-\pi, \pi] \times[-10,10]\). Use the graph to determine the following limits. a. \(\lim \limits_{x \rightarrow \pi / 2^{+}} \tan x\) b. \(\lim \limits_{x \rightarrow \pi / 2^{-}} \tan x\) c. \(\lim \limits_{x \rightarrow-\pi / 2^{+}} \tan x\) d. \(\lim \limits_{x \rightarrow-\pi / 2^{-}} \tan x\)

Graph the function y = sec x tan x with the window \([-\pi, \pi] \times[-10,10]\). Use the graph to determine the following limits. a. \(\lim \limits_{x \rightarrow \pi / 2^{+}} \sec x \tan x\) b. \(\lim \limits_{x \rightarrow \pi / 2^{-}} \sec x \tan x\) c. \(\lim \limits_{x \rightarrow-\pi / 2^{+}} \sec x \tan x\) d. \(\lim \limits_{x \rightarrow-\pi / 2^{-}} \sec x \tan x\)

Determine whether the following statements are true and give an explanation or counterexample. a. The line x = 1 is a vertical asymptote of the function \(f(x)=\frac{x^{2}-7 x+6}{x^{2}-1}\). b. The line x = -1 is a vertical asymptote of the function \(f(x)=\frac{x^{2}-7 x+6}{x^{2}-1}\). c. If g has a vertical asymptote at x = 1 and \(\lim _{x \rightarrow 1^{+}} g(x)=\infty\), then \(\lim \limits_{x \rightarrow 1^{-}} g(x)=\infty\).

Find polynomials p and q such that p/q is undefined at x = 1 and x = 2, but p/q has a vertical asymptote only at x = 2. Sketch a graph of your function.

Give a formula for a function f that satisfies \(\lim \limits_{x \rightarrow 6^{+}} f(x)=\infty\) and \(\lim \limits_{x \rightarrow 6^{-}} f(x)=-\infty\).

Match functions a-f with graphs A-F in the figure without using a graphing utility. a. \(f(x)=\frac{x}{x^{2}+1}\) b. \(f(x)=\frac{x}{x^{2}-1}\) c. \(f(x)=\frac{1}{x^{2}-1}\) d. \(f(x)=\frac{x}{(x-1)^{2}}\) e. \(f(x)=\frac{1}{(x-1)^{2}}\) f. \(f(x)=\frac{x}{x+1}\)

Use analytical methods and/or a graphing utility to identify the vertical asymptotes (if any) of the following functions. \(f(x)=\frac{x^{2}-3 x+2}{x^{10}-x^{9}}\)

Use analytical methods and/or a graphing utility to identify the vertical asymptotes (if any) of the following functions. \(g(x)=2-\ln x^{2}\)

Use analytical methods and/or a graphing utility to identify the vertical asymptotes (if any) of the following functions. \(h(x)=\frac{e^{x}}{(x+1)^{3}}\)

Use analytical methods and/or a graphing utility to identify the vertical asymptotes (if any) of the following functions. \(p(x)=\sec \left(\frac{\pi x}{2}\right),|x|<2\)

Use analytical methods and/or a graphing utility to identify the vertical asymptotes (if any) of the following functions. \(g(\theta)=\tan \left(\frac{\pi \theta}{10}\right)\)

Use analytical methods and/or a graphing utility to identify the vertical asymptotes (if any) of the following functions. \(q(s)=\frac{\pi}{s-\sin s}\)

Use analytical methods and/or a graphing utility to identify the vertical asymptotes (if any) of the following functions. \(f(x)=\frac{1}{\sqrt{x} \sec x}\)

Use analytical methods and/or a graphing utility to identify the vertical asymptotes (if any) of the following functions. \(g(x)=e^{1 / x}\)

Let \(f(x)=\frac{x^{2}-7 x+12}{x-a}\). a. For what values of a, if any, does \(\lim \limits_{x \rightarrow a^{+}} f(x)\) equal a finite number? b. For what values of a, if any, does \(\lim \limits_{x \rightarrow a^{+}} f(x)=\infty\)? c. For what values of a, if any, does \(\lim \limits_{x \rightarrow a^{+}} f(x)=-\infty\)?

a. Given the graph of f in the following figures, find the slope of the secant line that passes through (0,0) and (h, f(n)) in terms of h for h > 0 and h < 0. b. Calculate the limit of the slope of the secant line found in part (a) as \(h \rightarrow 0^{+}\) and \(h \rightarrow 0^{-}\). What does this tell you about the tangent line to the curve at (0, 0)? \(f(x)=x^{1 / 3}\)

a. Given the graph of f in the following figures, find the slope of the secant line that passes through (0,0) and (h, f(n)) in terms of h for h > 0 and h < 0. b. Calculate the limit of the slope of the secant line found in part (a) as \(h \rightarrow 0^{+}\) and \(h \rightarrow 0^{-}\). What does this tell you about the tangent line to the curve at (0, 0)? \(f(x)=x^{2 / 3}\)