Question Video: Finding the One-Sided Limit of a Function from Its Graph at a Point If the Limit Exists

Find lim_(๐‘ฅ โŸถ 5โป) ๐‘“(๐‘ฅ), if it exists.

02:10

Video Transcript

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

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

Since weโ€™re looking at the limit as ๐‘ฅ approaches five from the left and we know that our input values of ๐‘ฅ will be on the ๐‘ฅ-axis, letโ€™s mark ๐‘ฅ is equal to five. We can see something interesting about our function ๐‘“ of ๐‘ฅ 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, weโ€™re only interested in what happens as our values of ๐‘ฅ approach five. This means we donโ€™t need to worry what happens when ๐‘ฅ is equal to five. Weโ€™re only interested in what happens as ๐‘ฅ gets closer and closer to five, in this case, from the left.

So letโ€™s see what happens to our output values of ๐‘“ of ๐‘ฅ as ๐‘ฅ approaches five from the left. Thereโ€™s a few different ways of doing this. Letโ€™s start by inputting some values of ๐‘ฅ. Letโ€™s start when ๐‘ฅ is equal to two. We can see from the graph when ๐‘ฅ is equal to two, ๐‘“ of ๐‘ฅ outputs one, so ๐‘“ of two is equal to one. Letโ€™s now see what happens when ๐‘ฅ 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 ๐‘ฅ is equal to four. We get ๐‘“ of four is approximately equal to one-third.

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

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