# Estimating limit values from graphs

The best way to start reasoning about limits is using graphs. Learn how we analyze a limit graphically and see cases where a limit doesn't exist.
There's an important difference between the value a function is approaching—what we call the limit—and the value of the function itself. Graphs are a great tool for understanding this difference.
desmos.com to inspect $\displaystyle{\lim_{x \to 2}{\dfrac{x-2}{x^2-4}}}$
Notice how, as we get closer and closer to x=2 from both the left and the right, we seem to approach y=0.25.
In the example above, we see that the function value is undefined, but the limit value is approximately $0.25$.
Just remember that we're dealing with an approximation, not an exact value. We could zoom in further to get a better approximation if we wanted.

## Examples

The examples below highlight interesting cases of using graphs to approximate limits. In some of the examples, the limit value and the function value are equal, and in other examples, they are not.

### Sometimes the limit value equals the function value.

Problem 1
What is a reasonable estimate for $\displaystyle\lim_{x\to 1}g(x)$ ?
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### But, sometimes the limit value does not equal the function value.

Whenever you're dealing with a piecewise function, it's possible to get a graph like the one below.
Problem 2
What is a reasonable estimate for $\displaystyle\lim_{x\to 1}g(x)$ ?
The open circle says that the function doesn't have a value along the curve at $x=1$. And the closed, filled in circle is saying, "Hey, I'm the value of the function at $x=1$."
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Big takeaway: It's possible for the function value to be different from the limit value.

### And just because a function is undefined for some $x$-value doesn't mean there's no limit.

Holes in graphs happen with rational functions, which become undefined when their denominators are zero. Here's a classic example:
This is the graph of y = x / sin(x). Notice that there's a hole at x = 0 because the function is undefined there.
In this example, the limit appears to be $1$ because that's what the $y$-values seem to be approaching as our $x$-values get closer and closer to $0$. It doesn't matter that the function is undefined at $x=0$. The limit still exists.
Here's another problem for you to try:
Problem 3
What is a reasonable estimate for $\displaystyle\lim_{x\to -4}f(x)$ ?
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Reinforcing the key idea: The function value at $x=-4$ is irrelevant to finding the limit. All that matters is figuring out what the $y$-values are approaching as we get closer and closer to $x=-4$.

### On the flip side, when the function is defined for some $x$-value, that doesn't mean that the limit necessarily exists.

Just like an earlier example, this graph shows the sort of thing that can happen when we're working with piecewise functions. Notice how we're not approaching the same $y$-value from both sides of $x=3$.
Problem 4
What is a reasonable estimate for $\displaystyle\lim_{x\to 3}g(x)$?
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Want more practice? Try this exercise.

## Graphing calculators are pretty slick these days.

Graphing calculators like Desmos can give you a feel for what's happening to the $y$-values as you get closer and closer to a certain $x$-value. Try using a graphing calculator to estimate these limits:
\begin{aligned} &\displaystyle{\lim_{x \to 0}{\dfrac{x}{\sin(x)}}} \\\\ &\displaystyle{\lim_{x \to 3}{\dfrac{x-3}{x^2-9}}} \end{aligned}
In both cases, the function isn't defined at the $x$-value we're approaching, but the limit still exists, and we can estimate it.

## Summary questions

Problem 5
Is it always true that $\displaystyle{\lim_{x \to a}{f(x)}=f(a)}$?
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Problem 6
Which statement better describes how graphs help us reason about limits?
Choose all answers that apply:
Choose all answers that apply: