Question Video: Finding the Limit of a Function from Its Graph at a Point of Removable Discontinuity If the Limit Exists Mathematics • Higher Education

Determine the limit as π‘₯ ⟢ βˆ’1 of the function represented by the graph.

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Video Transcript

Determine the limit as π‘₯ approaches negative one of the function represented by the graph.

We’re given the graph of a function. And we need to determine the limit as π‘₯ approaches negative one of this function by using the graph. To start, we can see that our 𝑦-axis is labeled 𝑓 of π‘₯. So we’ll call our function 𝑓 of π‘₯. Let’s now recall what we mean by the limit as π‘₯ approaches negative one of a function 𝑓 of π‘₯.

The first thing we need to recall is the notation which represents this. We represent this by saying the limit as π‘₯ approaches negative one of 𝑓 of π‘₯. And what this means is, it’s the value that 𝑓 of π‘₯ approaches as π‘₯ tends to negative one. In other words, as our input values of π‘₯ are getting closer and closer to negative one, we want to see what happens to our output values 𝑓 of π‘₯.

Remember that our input values of π‘₯ will be on the π‘₯-axis. Since we want to know what happens as π‘₯ gets closer and closer to negative one, let’s mark this on our π‘₯-axis. And now we can see something interesting. We can see that our function 𝑓 of π‘₯ is not defined when π‘₯ is equal to negative one. This is represented by the hollow circle in our curve.

We might think this is a problem. However, remember, we’re only interested in what happens to our outputs as π‘₯ tends to negative one. This means we’re only interested in what happens to our outputs as π‘₯ gets closer and closer to negative one. Our value of π‘₯ will never be equal to negative one. We want to know what happens around this value.

Let’s start by seeing what happens as our values of π‘₯ approach negative one from the left. In other words, our inputs will all be less than negative one. To start, we can see if we input a value of π‘₯ is equal to negative six, then our function 𝑓 of π‘₯ outputs negative five. In other words, 𝑓 of negative six is negative five.

We can do the same when we input negative four. We can see that our function will output negative three. In other words, 𝑓 of negative four is equal to negative three. And we can continue doing this, getting closer. When we input negative two, our function outputs negative one. Well, we want to know what happens as our inputs are getting closer and closer to negative one. And if we continue doing this, we can see that our outputs are getting closer and closer to zero. But this is only one side of the story. What happens when our values of π‘₯ get closer and closer to negative one from the right? In other words, our inputs will all be bigger than negative one.

And we can answer this in exactly the same way. First, if we input six, we can see that our function outputs seven. Next, when our input value of π‘₯ is equal to four, we can see that our function outputs five. And we can keep doing this. When we input π‘₯ is equal to two, our function outputs three. And when we input π‘₯ is equal to zero, our function outputs one. And if we keep getting closer and closer to π‘₯ is equal to negative one, we can see that once again our outputs are approaching zero.

So in both cases, our outputs were getting closer and closer to zero. And because in both cases we were getting closer and closer to zero, we can conclude that this limit must be equal to zero. Therefore, we were able to show the limit as π‘₯ approaches negative one of the function 𝑓 of π‘₯ given to us in the graph is equal to zero.

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