### Video Transcript

Find the limit as π₯ approaches five from the left of π of π₯, if it exists.

Weβre given a sketch of the function π of π₯. We need to use it to determine the limit as π₯ approaches five from the left of π of π₯ if this limit exists. Letβs start by recalling what we mean by this limit. The limit as π₯ approaches five from the left of π of π₯ will be the value that π of π₯ approaches as π₯ tends to five and π₯ is less than five. In other words, as our input values of π₯ are getting closer and closer to five and π₯ approaches five from the left, we want to see what happens to our output values of π of π₯.

Since weβre looking at the limit as π₯ approaches five from the left and we know that our input values of π₯ will be on the π₯-axis, letβs mark π₯ is equal to five. We can see something interesting about our function π of π₯ at this point. The graph has a hollow circle. This means our function is undefined at this point. In other words, π of five does not exist. We might be worried about this, but, remember, weβre only interested in what happens as our values of π₯ approach five. This means we donβt need to worry what happens when π₯ is equal to five. Weβre only interested in what happens as π₯ gets closer and closer to five, in this case, from the left.

So letβs see what happens to our output values of π of π₯ as π₯ approaches five from the left. Thereβs a few different ways of doing this. Letβs start by inputting some values of π₯. Letβs start when π₯ is equal to two. We can see from the graph when π₯ is equal to two, π of π₯ outputs one, so π of two is equal to one. Letβs now see what happens when π₯ is equal to three. Either by calculating the gradients of our line or approximating, we can see that π of three is approximately equal to two-thirds. And we can do the same at π₯ is equal to four. We get π of four is approximately equal to one-third.

And we can keep doing this, taking values of π₯ getting closer and closer to five. And as we do this, we can see our outputs are getting closer and closer to zero. In other words, weβve shown as π₯ gets closer and closer to five from the left, our outputs π of π₯ are approaching zero. Therefore, by using this sketch, we were able to show the limit as π₯ approaches five from the left of π of π₯ is equal to zero.