Learning Materials
Features
Discover
Chapter 2: Problem 19
Determine the following limits. $$\lim _{x \rightarrow \infty}\left(3 x^{12}-9 x^{7}\right)$$
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
Limits with a parameter Let \(f(x)=\frac{x^{2}-7 x+12}{x-a}\) a. For what values of \(a,\) if any, does \(\lim _{x \rightarrow a^{+}} f(x)\) equal a finite number? b. For what values of \(a,\) if any, does \(\lim _{x \rightarrow a^{+}} f(x)=\infty ?\) c. For what values of \(a,\) if any, does \(\lim _{x \rightarrow a^{+}} f(x)=-\infty ?\)
The limit at infinity \(\lim _{x \rightarrow \infty} f(x)=L\) means that for any \(\varepsilon>0,\) there exists \(N>0\) such that $$|f(x)-L|<\varepsilon \quad \text { whenever } \quad x>N$$ Use this definition to prove the following statements. $$\lim _{x \rightarrow \infty} \frac{10}{x}=0$$
If a function \(f\) represents a system that varies in time, the existence of \(\lim _{t \rightarrow \infty} f(t)\) means that the system reaches a steady state (or equilibrium). For the following systems, determine if a steady state exists and give the steady-state value. The amount of drug (in milligrams) in the blood after an IV tube is inserted is \(m(t)=200\left(1-2^{-t}\right)\).
Explain why or why not 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 \(\lim _{x \rightarrow 1^{-}} g(x)=\infty.\)
Evaluate \(\lim _{x \rightarrow \infty} f(x)\) and \(\lim _{x \rightarrow-\infty} f(x)\). Then state the horizontal asymptote(s) of \(f\). Confirm your findings by plotting \(f\). $$f(x)=\frac{3 e^{x}+e^{-x}}{e^{x}+e^{-x}}$$
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