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Chapter 2: Problem 11

Evaluate the following limits. $$\lim _{\theta \rightarrow \infty} \frac{\cos \theta}{\theta^{2}}$$

Short Answer

Expert verified
Question: Determine the limit of the function \(\frac{\cos\theta}{\theta^2}\) as 胃 approaches infinity. Answer: The limit of the function \(\frac{\cos\theta}{\theta^2}\) as 胃 approaches infinity is 0.

Step by step solution

01

Analyze the cosine function

Cosine is a periodic function with a period of \(2\pi\), and its value ranges from -1 to 1, i.e., \(-1 \leq \cos \theta \leq 1\). As 胃 approaches infinity, the value of the cosine function continues to oscillate between -1 and 1, but it does not approach any fixed value.
02

Analyze the denominator behavior as 胃 approaches infinity

As 胃 approaches infinity, the square of 胃 (that is, \(\theta^2\)) goes to infinity as well. So, the denominator of our given fraction becomes infinitely large.
03

Evaluate the limit

Since the numerator of our fraction (\(\cos \theta\)) oscillates between -1 and 1, and the denominator (\(\theta^2\)) goes to infinity, the overall fraction approaches 0 as 胃 approaches infinity. Therefore, we can state that: $$\lim_{\theta \rightarrow \infty} \frac{\cos\theta}{\theta^2} = 0$$

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Most popular questions from this chapter

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. $$\lim _{x \rightarrow \infty} \frac{10}{x}=0$$

Creating functions satisfying given limit conditions Find a function \(f\) satisfying \(\lim _{x \rightarrow 1}\left(\frac{f(x)}{x-1}\right)=2\).

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

a. Find functions \(f\) and \(g\) such that each function is continuous at \(0\), but \(f \circ g\) is not continuous at \(0 .\) b. Explain why examples satisfying part (a) do not contradict Theorem 11.

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 \(\lim _{x \rightarrow 1^{-}} g(x)=\infty.\)
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