Chapter 2: Problem 47
Asymptotes Use analytical methods and/or a graphing utility to identify the vertical asymptotes (if any) of the following functions. $$h(x)=\frac{e^{x}}{(x+1)^{3}}$$
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Find polynomials \(p\) and \(q\) such that \(p / q\) is undefined at 1 and \(2,\) but \(p / q\) has a vertical asymptote only at \(2 .\) Sketch a graph of your function.
Evaluate the following limits. $$\lim _{x \rightarrow 3 \pi / 2} \frac{\sin ^{2} x+6 \sin x+5}{\sin ^{2} x-1}$$
Theorem 4a Given the polynomial $$p(x)=b_{n} x^{n}+b_{n-1} x^{n-1}+\cdots+b_{1} x+b_{0}$$ prove that \(\lim _{x \rightarrow a} p(x)=p(a)\) for any value of \(a\)
a. Evaluate \(\lim _{x \rightarrow \infty} f(x)\) and \(\lim _{x \rightarrow-\infty} f(x),\) and then identify any horizontal asymptotes. b. Find the vertical asymptotes. For each vertical asymptote \(x=a\), evaluate \(\lim _{x \rightarrow a^{-}} f(x)\) and \(\lim _{x \rightarrow a^{+}} f(x)\). $$f(x)=\frac{x^{2}-9}{x(x-3)}$$
Prove Theorem 11: If \(g\) is continuous at \(a\) and \(f\) is continuous at \(g(a),\) then the composition \(f \circ g\) is continuous at \(a .\) (Hint: Write the definition of continuity for \(f\) and \(g\) separately; then, combine them to form the definition of continuity for \(\left.f^{\circ} g .\right)\)
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