§1.7 Limits, continuity, and differentiability

Introduction

Having a limit at a point

Being continuous at a point

Being differentiable at a point

A function $f$ is given by the graph in the following figure. Use the graph to answer each of the following questions.

Note: to the right of $x = 2$, the graph of $f$ is exhibiting infinite oscillatory behavior similar to the function $\sin(\frac{\pi}{x})$. Assume that $f(2) = -2.5$.

For each of the values $a = -3, -2, -1, 0, 1, 2, 3$, determine whether or not $\lim_{x \to a} f(x)$ exists. If the function has a limit $L$ at a given $x$-value, state the value of the limit using the notation $\lim_{x \to a} f(x) = L$. If the function does not have a limit at a given point, write a sentence to explain why.

For each of the values $a = -2, -1, 0, 1, 2$, compute $f(a)$.

For each of the values $a = -2, -1, 0, 1, 2$, determine $\displaystyle \lim_{x \to a^-} f(x)$ and $\displaystyle \lim_{x \to a^+} f(x)$.

For each of the values $a = -2, -1, 0, 1, 2$, determine $\displaystyle \lim_{x \to a} f(x)$. If the limit fails to exist, explain why by discussing the left- and right-hand limits at the relevant $a$-value.

For which values of $a$ is the following statement true?

On the axes provided, sketch an accurate, labeled graph of $y = f(x)$. Be sure to carefully use open circles (○) and filled circles (●) to represent key points on the graph, as dictated by the piecewise formula.

At which values of $a$ does $\lim_{x \to a} f(x)$ not exist?

At which values of $a$ is $f(a)$ not defined?

At which values of $a$ does $f$ have a limit, but $\lim_{x \to a} f(x) \ne f(a)$?

State all values of $a$ for which $f$ is not continuous at $x = a$.

Which condition is stronger, and hence implies the other: $f$ has a limit at $x = a$ or $f$ is continuous at $x = a$? Explain, and hence complete the following sentence: “If $f$ at $x = a$, then $f$ at $x = a$,” where you complete the blanks with has a limit and is continuous, using each phrase once.

Reasoning graphically (that is, by discussing the graph of $g(x)=|x|$), explain why $g$ is differentiable at every point $x$ such that $x \ne 0$.

Use the limit definition of the derivative to show that $g'(0) = \lim_{h \to 0} \frac{|h|}{h}$.

Explain why $g'(0)$ fails to exist by using small positive and negative values of $h$.

Let $f$ be the function that we have previously explored in Preview Activity 1.7, whose graph is given again in the following figure.

State all values of $a$ for which $f$ is not differentiable at $x = a$. For each, provide a reason for your conclusion.

True or false: if a function $p$ is differentiable at $x = b$, then $\lim_{x \to b} p(x)$ must exist. Why?