TL5 Graphical Limits - MTH 121 Spring 2026

Summary So Far¶

Up until this point we’ve seen two ways to find the tangent line:

Estimate it, by drawing a line that roughly points in the same direction as the curve.
Use the power rule to get the slope and then find the equation of the line.

The question remains, where does this power rule come from? Or more generally, how do we construct the tangent line precisely, so that it is not an estimate. To answer these questions we need to introduce the idea of limits.

Definition of Limit¶

Uses of Limits¶

Two of the primary uses we will see for limits are:

Using limits to help describe the behavior of a function near a specific $x$ value.
Using limits to look at small movements along a curve.

One-Sided Limits¶

Since the $x$ axis is a 1 dimensional object, there are only two directions that we can approach a specific $x=a$ value: from the left side and from the right side.

Left Hand Limit

\lim_{x\to a^-} f(x) = L

Right Hand Limit

\lim_{x\to a^+} f(x) = L

If both of these limits exist and are equal to each other, then the overall limit exists and is equal to the shared value.

Primary Examples¶

Example 1

The graph of function $f(x)$ is given below. Use this graph to calculate the following:

Behavior near $x=1$	Behavior near $x=4$
$\displaystyle \lim_{x\to 1^-} f(x)$	$\displaystyle \lim_{x\to 4^-} f(x)$
$\displaystyle \lim_{x\to 1^+} f(x)$	$\displaystyle \lim_{x\to 4^+} f(x)$
$\displaystyle \lim_{x\to 1} f(x)$	$\displaystyle \lim_{x\to 4} f(x)$
$\displaystyle f(1)$	$\displaystyle f(4)$

Example 2

The graph of function $f(x)$ is given below. Use this graph to calculate the following:

Behavior near $x=1$	Behavior near $x=2$
$\displaystyle \lim_{x\to -1^-} f(x)$	$\displaystyle \lim_{x\to 2^-} f(x)$
$\displaystyle \lim_{x\to -1^+} f(x)$	$\displaystyle \lim_{x\to 2^+} f(x)$
$\displaystyle \lim_{x\to -1} f(x)$	$\displaystyle \lim_{x\to 2} f(x)$
$\displaystyle f(-1)$	$\displaystyle f(2)$

Homework¶

Coming soon!

Write out each question as well as your solution on the homework page template.

Make sure you leave a good amount of blank space between each question.
Write down the question as well as your solution.