§1.7 Limits, continuity, and differentiability

Introduction

Having a limit at a point

Being continuous at a point

Being differentiable at a point

A graph has a hole with matching sides at $x = -3$, a jump at $x = -2$, a corner at $x = 0$, smooth continuous passes at $x = -1$, $1$, and $3$, and infinite oscillation like $\sin(\frac{\pi}{x})$ just right of $x = 2$. At which inputs does $\lim_{x \to a} f(x)$ fail to exist?

A piecewise function has $f(-2) = 1$, $f(-1)$ undefined, $f(0) = \frac{7}{3}$, $f(1) = 2$, $f(2) = 2$. Its one-sided limits are: at $-2$, left $2$ and right $1$; at $-1$, both sides $\frac{5}{3}$; at $0$, both sides $\frac{7}{3}$; at $1$, both sides $3$; at $2$, both sides $2$. Which statement is correct?

For the same piecewise function, at which listed inputs does $\lim_{x \to a} f(x) \ne f(a)$?

When sketching a graph from a piecewise formula, an endpoint that a piece excludes is drawn as an open circle, and if the neighboring piece actually attains that height, it supplies the filled circle at the same point.

A graph shows $f$ undefined only at $a = 3$, while $\lim_{x \to 3} f(x) = -2.5$; at $a = -1$ the graph passes near height $-3.25$ on both sides but the filled point sits at $(-1, 1)$. Where does $f$ have a limit with $\lim_{x \to a} f(x) \ne f(a)$?

For that graph (jump at $-2$, oscillation at $2$, undefined at $3$, mismatched value at $-1$), at which inputs is $f$ not continuous?

If $f$ is continuous at $x = a$, then $f$ has a limit at $x = a$.

For $g(x) = |x|$, compute the derivative value $g'(-3)$ on the left branch of the graph.

Compute $\displaystyle\lim_{h \to 0^+} \frac{|h|}{h}$.

Let $g(x) = |x|$. Which statement explains why $g'(0)$ does not exist?

A graph is discontinuous at $a = -2, -1, 2, 3$ and has corner points at $a = -3$ and $a = 1$. At which inputs is $f$ not differentiable, and why?

If a function $p$ is differentiable at $x = b$, which conclusion must hold?