Graficos de limites com duas variaveis

(Videos) Finding Limits from a Graph

by PatrickJMT

In this video by patrickJMT he shows you how look at a graph an evaluate a limit.

He explains this idea by looking at an arbitrary graph which has a break in it at the $x$ value of $3$.

If we take an $x$ value that gets closer and closer to $3$, whether it's from the left of $3$ or the right of $3$, what are the $y$ values getting closer to?

As $x$ gets closer to $3$, $f(x)$ (the $y$ values) gets closer to $2$.

Even though $f(x)$ is undefined at $2$ because of the open circle, the graph still gets closer and closer to that value as we move closer to $3$. We can write this in mathematical terms using a limit. $$\lim_{x\rightarrow3}f(x)=2$$ All this says is if you have some function $f(x)$ and you plug in $x$ values close to $3$ you get out values close to $2$.

He then looks at a graph and starts answering some questions. It looks similar to:

The first question he looks at is: what is $f(0)$? All we do is plug in the $x$ value of $0$ and figure out what the corresponding $y$ value is. So we can write: $$f(0)=2$$

The next question is: what is $\lim_{x\rightarrow0}f(x)$? What we need to do is look at the values on both sides of $x=0$ as they approach it. Then look at their corresponding $y$ values and see if they get closer to a number. If we look at our graph, we can see they get closer to $2$. So we can write: $$\lim_{x\rightarrow0}f(x)=2$$

Next up is: what is $f(-2)$? If we plug in $x=-2$, since our dot is up at $5$, we get out that. So: $$f(-2)=5$$

Following that we have: what is $\lim_{x\rightarrow-2}f(x)$? Well, as we get closer to $x=-2$ from both sides, the graph gets closer to the $y$ value of $1$. So: $$\lim_{x\rightarrow-2}f(x)=1$$

The next problem is: what is $\lim_{x\rightarrow2}f(x)$? In this case, as the $x$ values approach $2$ the $y$ values get bigger and bigger. So we say that the limit approaches infinity, or: $$\lim_{x\rightarrow2}f(x)=\infty$$

Then he asks: what is $f(4)$? If we plug $4$ in we get out the $y$ value $2$. So: $$f(4)=2$$

What if we were asked to find $\lim_{x\rightarrow4}f(x)$? If we approach from the left, that is take $x$ points smaller than $4$, the limit approaches $2$. But if we approach from the right, that is take $x$ points larger than $4$, the limit approaches $4$. Since these are two separate numbers that the limit is approaching, we say the limit does not exist. $$\lim_{x\rightarrow4}f(x)\text{ DNE}$$

However, we can talk about limits specifically from the left or right. If you were asked to find $\lim_{x\rightarrow4^-}f(x)$ (there's a little negative sign above the 4) we would look at the $x$ values as they approach from the left, or values smaller than 4.

If you were asked to find $\lim_{x\rightarrow4^+}f(x)$ (there's a little positive sign above the 4) we would look at the $x$ values as they approach from the right, or values larger than 4.

So: $$\lim_{x\rightarrow4^-}f(x)=2\\ \lim_{x\rightarrow4^+}f(x)=4$$

In this video he looks at another graph and asks some questions about it regarding limits.

The graph he looks at is:

See if you can answer these questions on your own!

(Video) Finding a Limit of a Piecewise Function by Graphing

by PatrickJMT

In this video by patrickJMT he shows you how you can answer questions about limits by graphing.

The problem he looks at is:

Let $g(x)=\begin{cases}x^2+1;&x<0\\2x+3;&x\geq0\end{cases}$. What is the limit of $g(x)$ as $x$ approaches $-2$?

So one way to solve this type of problem is to simply just graph it. So let's do just that.

So what we want to evaluate is: $$\lim_{x\rightarrow-2}g(x)$$

So if we look at our graph and approach the $x$ value $-2$ we see that it approaches the $y$ value of $5$. We can also plug $x=-2$ into our function. $$g(-2)=(-2)^2+1\\ g(-2)=4+1\\ g(-2)=5$$ So we would just write $$\lim_{x\rightarrow-2}f(x)=5$$

If we were asked to evaluate $\lim_{x\rightarrow0}f(x)$ we could graph the function and see that each side approaches a different number. This tells us that the limit does not exist. $$\lim_{x\rightarrow0}f(x)\text{ DNE}$$