### Video Transcript

Find the limit as ๐ฅ approaches five from the left of ๐ of ๐ฅ, if it exists.

Weโre given a sketch of the function ๐ of ๐ฅ. We need to use it to determine the limit as ๐ฅ approaches five from the left of ๐ of ๐ฅ if this limit exists. Letโs start by recalling what we mean by this limit. The limit as ๐ฅ approaches five from the left of ๐ of ๐ฅ will be the value that ๐ of ๐ฅ approaches as ๐ฅ tends to five and ๐ฅ is less than five. In other words, as our input values of ๐ฅ are getting closer and closer to five and ๐ฅ approaches five from the left, we want to see what happens to our output values of ๐ of ๐ฅ.

Since weโre looking at the limit as ๐ฅ approaches five from the left and we know that our input values of ๐ฅ will be on the ๐ฅ-axis, letโs mark ๐ฅ is equal to five. We can see something interesting about our function ๐ of ๐ฅ at this point. The graph has a hollow circle. This means our function is undefined at this point. In other words, ๐ of five does not exist. We might be worried about this, but, remember, weโre only interested in what happens as our values of ๐ฅ approach five. This means we donโt need to worry what happens when ๐ฅ is equal to five. Weโre only interested in what happens as ๐ฅ gets closer and closer to five, in this case, from the left.

So letโs see what happens to our output values of ๐ of ๐ฅ as ๐ฅ approaches five from the left. Thereโs a few different ways of doing this. Letโs start by inputting some values of ๐ฅ. Letโs start when ๐ฅ is equal to two. We can see from the graph when ๐ฅ is equal to two, ๐ of ๐ฅ outputs one, so ๐ of two is equal to one. Letโs now see what happens when ๐ฅ is equal to three. Either by calculating the gradients of our line or approximating, we can see that ๐ of three is approximately equal to two-thirds. And we can do the same at ๐ฅ is equal to four. We get ๐ of four is approximately equal to one-third.

And we can keep doing this, taking values of ๐ฅ 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, weโve shown as ๐ฅ gets closer and closer to five from the left, our outputs ๐ of ๐ฅ are approaching zero. Therefore, by using this sketch, we were able to show the limit as ๐ฅ approaches five from the left of ๐ of ๐ฅ is equal to zero.