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Chapter 4: Problem 3
Find \(\lim _{x \rightarrow 0} \frac{\sqrt{1+2 x}-\sqrt{1+3 x}}{x+2 x^{2}}\) where \(x>0\)
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Suppose that \(f\) and \(g\) have limits in \(\mathbb{R}\) as \(x \rightarrow \infty\) and that \(f(x) \leq g(x)\) for all \(x \in(a, \infty)\). Prove that \(\lim _{x \rightarrow \infty} f \leq \lim _{x \rightarrow \infty} g\).
Show that the following limits do not exist. (a) \(\lim _{x \rightarrow 0} \frac{1}{x^{2}} \quad(x>0)\), (b) \(\lim _{x \rightarrow 0} \frac{1}{\sqrt{x}} \quad(x>0)\), (c) \(\lim _{x \rightarrow 0}(x+\operatorname{sgn}(x))\), (d) \(\lim _{x \rightarrow 0} \sin \left(1 / x^{2}\right)\).
Let \(I\) be an interval in \(\mathbb{R}\). let \(f: I \rightarrow \mathbb{R}\), and let \(c \in I\). Suppose there exist constants \(K\) and \(L\) such that \(|f(x)-L| \leq K|x-c|\) for \(x \in I .\) Show that \(\lim _{x \rightarrow c} f(x)=L\)
Give an example of a function that has a right-hand limit but not a left-band limit at a point.
Let \(c \in \mathbb{R}\) and let \(f\) be defined for \(x \in(c, \infty)\) and \(f(x)>0\) for all \(x \in(c, \infty)\). Show that \(\lim _{x \rightarrow c} f=\infty\) if and only if \(\lim _{x \rightarrow c} 1 / f=0\)
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