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

Question Video: Finding the One-Sided Limit of a Function from Its Graph at a Point If the Limit Exists Mathematics • Second Year of Secondary School

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Find lim_(π‘₯ ⟢ 5⁻) 𝑓(π‘₯), if it exists.

02:10

Video Transcript

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

We’re given a sketch of the function 𝑓 of π‘₯. We need to use it to determine the limit as π‘₯ approaches five from the left of 𝑓 of π‘₯ if this limit exists. Let’s start by recalling what we mean by this limit. The limit as π‘₯ approaches five from the left of 𝑓 of π‘₯ will be the value that 𝑓 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 𝑓 of π‘₯.

Since we’re 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, let’s mark π‘₯ is equal to five. We can see something interesting about our function 𝑓 of π‘₯ at this point. The graph has a hollow circle. This means our function is undefined at this point. In other words, 𝑓 of five does not exist. We might be worried about this, but, remember, we’re only interested in what happens as our values of π‘₯ approach five. This means we don’t need to worry what happens when π‘₯ is equal to five. We’re only interested in what happens as π‘₯ gets closer and closer to five, in this case, from the left.

So let’s see what happens to our output values of 𝑓 of π‘₯ as π‘₯ approaches five from the left. There’s a few different ways of doing this. Let’s start by inputting some values of π‘₯. Let’s start when π‘₯ is equal to two. We can see from the graph when π‘₯ is equal to two, 𝑓 of π‘₯ outputs one, so 𝑓 of two is equal to one. Let’s now see what happens when π‘₯ is equal to three. Either by calculating the gradients of our line or approximating, we can see that 𝑓 of three is approximately equal to two-thirds. And we can do the same at π‘₯ is equal to four. We get 𝑓 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, we’ve shown as π‘₯ gets closer and closer to five from the left, our outputs 𝑓 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 𝑓 of π‘₯ is equal to zero.

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