Limit Meaning Practice Problems

In the statement $\lim_{x\to4}f(x)=9$, identify the input target and output target.

Write in symbols: "The limit of $g(t)$ as $t$ approaches $2$ is $-5$."

Write in words: $\lim_{h\to0}Q(h)=12$.

True or false: if $\lim_{x\to a}f(x)=L$, then $f(a)=L$. Explain.

True or false: a function must be defined at $a$ for $\lim_{x\to a}f(x)$ to exist. Explain.

What does the superscript minus mean in $x\to3^-$?

What does the superscript plus mean in $x\to3^+$?

Explain why $x\to2$ does not mean $x=2$.

A graph approaches $7$ from both sides at $x=1$, but $f(1)=10$. Find the limit.

A graph has no filled point at $x=-2$, but both sides approach $4$. Find $f(-2)$ and the limit.

A car's position is $s(t)=5t+2$. Find its average velocity on $[1,4]$.

A particle's position is $s(t)=t^2$. Find its average velocity on $[3,3+h]$, simplify, and predict the instantaneous velocity at $t=3$.

A ball's height is $s(t)=80t-16t^2$. Find its average velocity on $[2,2+h]$ and its instantaneous velocity at $t=2$.

A population model is $P(t)=1000+50t+2t^2$. Find the average rate of change on $[4,4+h]$ and predict $P'(4)$ informally from the limit.

Explain geometrically what the average rate of change represents on a graph.

Explain geometrically what the limiting secant line becomes when the limit exists.

Complete a table near $x=3$ for $f(x)=\dfrac{x^2-9}{x-3}$, then estimate the limit.

Complete a table near $x=2$ for $g(x)=\dfrac{x^3-8}{x-2}$, then estimate the limit.

For $h(x)=\dfrac{x^2+x-6}{x-2}$, simplify for $x\ne2$ and find the limit at $2$.

For $p(x)=\dfrac{x^2-1}{x-1}$, find $p(1)$ and $\lim_{x\to1}p(x)$.

Define $q(1)=100$ and $q(x)=x+1$ for $x\ne1$. Find $q(1)$ and the limit at $1$.

Create two functions with different values at $x=0$ but the same limit as $x\to0$.

Let $f(x)=\begin{cases}2x+1,&x<1,\\x+3,&x\ge1.\end{cases}$ Find both one-sided limits and the two-sided limit at $1$.

Let $g(x)=\begin{cases}x^2,&x\le2,\\3x-1,&x>2.\end{cases}$ Find both one-sided limits at $2$.

Let $h(x)=\begin{cases}4,&x<0,\\4+x,&x\ge0.\end{cases}$ Does the limit at $0$ exist?

Let $k(x)=\begin{cases}-1,&x<3,\\2,&x\ge3.\end{cases}$ Explain why the limit at $3$ does not exist.

Find $\lim_{x\to0^-}|x|/x$ and $\lim_{x\to0^+}|x|/x$.

Find $\lim_{x\to2^-}|x-2|/(x-2)$ and the corresponding right-hand limit.

State the reason $\lim_{x\to0}1/x$ does not exist as a real number.

State the reason $\lim_{x\to0}\sin(1/x)$ does not exist.

Does $\lim_{x\to0^+}\sqrt{x}$ exist? Does the ordinary two-sided limit exist in the real domain?

A student says, "The limit is $7$ because the filled dot is at $(2,7)$." What information is missing?

A student says, "The limit does not exist because the function is undefined at the point." Give a counterexample.

Construct a function whose left-hand limit at $1$ is $2$, right-hand limit is $5$, and function value is $-3$.

Construct a function that is undefined at $4$ but has limit $9$ there.

Explain why a finite table can suggest but cannot prove the existence of a limit for every possible function.

Suppose

Find both one-sided limits, the two-sided limit, and $f(2)$.

Suppose $f(x)=x+4$ for every $x\ne-1$, and $f(-1)$ is not given. Determine the limit and list every possible value of $f(-1)$ consistent with that limit.

A moving object's average velocity from $t=5$ to $t=5+h$ simplifies to $18-3h$. Find its instantaneous velocity at $t=5$, and explain the role of the limit.

Give a graph description for a function satisfying

Give a graph description for a function satisfying

Explain why the first function in the previous problem has a two-sided limit and the second does not.