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Chapter 2: Problem 23

Use the precise definition of a limit to prove the following limits. $$\left.\lim _{x \rightarrow 0} x^{2}=0 \text { (Hint: Use the identity } \sqrt{x^{2}}=|x| .\right)$$

Short Answer

Expert verified
Answer: The delta value chosen was \(\delta = \sqrt{\epsilon}\).

Step by step solution

01

Recall the Precise Definition of a Limit

The precise definition of a limit states that: For a given function \(f(x)\), the limit as \(x \rightarrow a\) is \(L\) if for any \(\epsilon > 0\), there exists \(\delta > 0\) such that if \(0 < |x - a| < \delta\), then \(|f(x) - L| < \epsilon\). In our case, we want to prove that \(\lim_{x \rightarrow 0} x^2 = 0\). To do this, we need to find a delta for the given epsilon and then show that the inequalities hold.
02

Set up the Inequality

We need to show that for any \(\epsilon > 0\), there exists \(\delta > 0\) such that if \(0 < |x - 0| < \delta\), then \(|x^2 - 0| < \epsilon\). We can rewrite the inequalities as follows: 1. \(0 < |x| < \delta\) 2. \(|x^2| < \epsilon\)
03

Taking Hint into Consideration

Using the hint given in the exercise, we know that \(\sqrt{x^2} = |x|\). We can take advantage of this and further simplify the second inequality. So the inequality can be written as: $$|\sqrt{x^2}| < \epsilon$$
04

Finding an Appropriate Delta

Now, we need to find an appropriate delta in terms of epsilon. Since we have \(|\sqrt{x^2}| < \epsilon\), we can easily say that if \(|x| < \sqrt{\epsilon}\), then \(|x|^2 = x^2 < \epsilon\). So, we can choose \(\delta = \sqrt{\epsilon}\).
05

Putting It All Together

We have found that if \(\delta = \sqrt{\epsilon}\), the inequality holds. Therefore, for any given \(\epsilon > 0\), there exists \(\delta > 0\) (in this case, \(\delta = \sqrt{\epsilon}\)), such that if \(0 < |x| < \delta\), then \(|x^2| < \epsilon\). This completes the proof, and we have now shown that \(\lim_{x \rightarrow 0} x^2 = 0\) using the precise definition of a limit.

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Most popular questions from this chapter

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 \(\lim _{x \rightarrow a^{+}} f(x)\). $$f(x)=\frac{x-1}{x^{2 / 3}-1}$$

Find the vertical and horizontal asymptotes of \(f(x)=e^{1 / x}\).

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 \(\lim _{x \rightarrow a^{+}} f(x)\). $$f(x)=16 x^{2}\left(4 x^{2}-\sqrt{16 x^{4}+1}\right)$$

Determine the end behavior of the following transcendental functions by evaluating appropriate limits. Then provide a simple sketch of the associated graph, showing asymptotes if they exist. $$f(x)=2^{x}$$

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