Learning Materials
Features
Discover
Chapter 2: Problem 7
What is the domain of \(f(x)=e^{x} / x\) and where is \(f\) continuous?
Over 30 million students worldwide already upgrade their learning with 91影视!
All the tools & learning materials you need for study success - in one app.
Get started for free
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 \(\left.f^{\circ} g .\right)\)
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 \(\lim _{x \rightarrow a^{+}} f(x)\). $$f(x)=\frac{\sqrt{x^{2}+2 x+6}-3}{x-1}$$
Investigate the following limits. $$\lim _{\theta \rightarrow \pi / 2^{+}} \frac{1}{3} \tan \theta$$
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 \(\lim _{x \rightarrow a^{+}} f(x)\). $$f(x)=16 x^{2}\left(4 x^{2}-\sqrt{16 x^{4}+1}\right)$$
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 \(\lim _{x \rightarrow a^{+}} f(x)\). $$f(x)=\frac{3 x^{4}+3 x^{3}-36 x^{2}}{x^{4}-25 x^{2}+144}$$
What do you think about this solution?
We value your feedback to improve our textbook solutions.
Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine