Estimate limit values from graphs

▶ Estimating limit values from graphs

In this video, we learn to estimate limit values from graphs by observing the function's behavior as x approaches a value from both left and right sides. If the function approaches the same value from both sides, the limit exists. If it approaches different values or is unbounded, the limit doesn't exist.

⏱ 6:01 💾 5.8 MB License: cc-by-nc-sa Streaming via CDN

▶ Examining unbounded limits

This video discusses estimating limit values from graphs, focusing on two functions: y = 1/x² and y = 1/x. For y = 1/x², the limit is unbounded as x approaches 0, since the function increases without bound. For y = 1/x, the limit doesn't exist as x approaches 0, since it's unbounded in opposite directions.

⏱ 2:32 💾 3.6 MB License: cc-by-nc-sa Streaming via CDN

📄 Estimating limit values from graphs

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.

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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.

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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.

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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:

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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:

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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 $x=3$.

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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:

$\begin{align}
&\displaystyle{\lim_{x \to 0}{\dfrac{x}{\sin(x)}}}
\\
&\displaystyle{\lim_{x \to 3}{\dfrac{x-3}{x^2-9}}}
\end{align}$

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

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▶ Estimating one-sided limits from graphs

A one-sided limit is the value the function approaches as the x-values approach the limit from *one side only*. For example, f(x)=|x|/x returns -1 for negative numbers, 1 for positive numbers, and isn't defined for 0. The one-sided *right* limit of f at x=0 is 1, and the one-sided *left* limit at x=0 is -1.

⏱ 9:11 💾 9.0 MB License: cc-by-nc-sa Streaming via CDN

▶ Estimating one-sided limits from graphs: asymptote

This video explores estimating one-sided limit values from graphs. As x approaches 6 from the left, the function becomes unbounded with an asymptote, making the left-sided limit nonexistent. However, when approaching 6 from the right, the function approaches -3, indicating that the right-handed limit exists. Sal's analysis highlights the importance of understanding limits from both sides.

⏱ 2:27 💾 5.1 MB License: cc-by-nc-sa Streaming via CDN

▶ Connecting limits and graphical behavior

Usually when we analyze a function's limits from its graph, we are looking at the more "interesting" points. It's important to remember that you can talk about the function's value at any point. Also, a description of a limit can apply to multiple different functions.

⏱ 5:03 💾 7.6 MB License: cc-by-nc-sa Streaming via CDN

▶ Missing function detail when zoomed out

Understand because of issues of scale, graphical representations of functions may miss important function behavior.

⏱ 3:29 💾 4.9 MB License: cc-by-nc-sa Streaming via CDN

▶ Missing function features when zoomed in

Understand because of issues of scale, graphical representations of functions may miss important function behavior.

⏱ 3:18 💾 5.0 MB License: cc-by-nc-sa Streaming via CDN

▶ Finding limit of an oscillating function

A limit might not exist if the function is oscillating near a value of x.

⏱ 1:15 💾 1.7 MB License: cc-by-nc-sa Streaming via CDN