Continuous Functions Exercise 1 For which functions $ f $ does there exist a continuous function $ F: \mathbb{R} \to \mathbb{R} $ such that $ F(x) = f(x) $ for all $ x $ in the domain of $ f $? $ f(x) = \frac{x^2 - 4}{x - 2} $ $ f(x) = \frac{|x|}{x} $ $ f(x) = 0 $ for irrational $ x $ $ f(x) = \frac{1}{q} $ for $ x = \frac{p}{q} $ in lowest terms ...
Limits and Continuity Exercise 1 Find the following limits: $\lim_{x \to 1} \frac{x^2 - 1}{x^3 - 1}$ $\lim_{x \to 2} \frac{x^3 - 8}{x^2 - 4}$ $\lim_{x \to 3} \frac{x^3 - 8}{x^2 - 4}$ (Note the changed limit point) $\lim_{x \to y} \frac{x^n - y^n}{x - y}$ $\lim_{y \to x} \frac{x^n - y^n}{x - y}$ $\lim_{h \to 0} \frac{\sqrt{a + h} - \sqrt{a}}{h}$ Exercise 2 Find the following limits: $\lim_{x \to 1} \frac{1 - \sqrt{x}}{1 - x}$ $\lim_{x \to 0} \frac{1 - \sqrt{1 - x^2}}{x}$ $\lim_{x \to 0} \frac{1 - \sqrt{1 - x^2}}{x^2}$ Exercise 3 In each of the following cases, determine the limit $l$ for the given $a$, and prove that it is the limit by showing how to find a $\delta$ such that $| f(x) - l | < \varepsilon$ where $\forall x$ satisfying $0 < |x-a| < \delta$ ...