Homework 11: Limits of Functions

Instructions. Assignments should be stapled and written clearly and legibly. For problems 1, 3, and 4, you must use the \(\epsilon-\delta\) definition of a limit. Problem 19 from the textbook is optional.

1. Section 5.1, #4, 7(a), 13 (the Squeeze Theorem), 19.
2. Use the Squeeze Theorem to prove that \( \lim\limits_{x\to 0} \left(x \sin\left(\frac{1}{x}\right)\right) = 0\). Make sure to state what \(D\) is.
3. Let \(f:D\to \bR\), where \(D\subseteq \bR\), and let \(a\) be a limit point of \(D\). Suppose that \(\lim\limits_{x\to a}f(x) \gt 0\). Prove that there exists a deleted neighborhood \(N_{\delta}^*(a)\) of \(a\) such that \(f(x)\gt 0\) for all \(x\in N_{\delta}^*(a)\cap D\).
4. Let \(a \gt 0\). Use the definition of limit to prove that \( \lim\limits_{x\to a}\sqrt{x}=\sqrt{a}\).
   Hint: Use the inequality \( |\sqrt{x}-\sqrt{a}|=\dfrac{|x-a|}{\sqrt{x}+\sqrt{a}}\leq \dfrac{|x-a|}{\sqrt{a}}\)