# Question Video: Finding the Limit of a Function from Its Graph Mathematics • Higher Education

If the graph shown represents the function π(π₯) = π₯ β 3, determine lim_(π₯ β β1) π(π₯).

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### Video Transcript

If the graph shown represents the function π of π₯ equals π₯ minus three, determine the limit as π₯ tends to negative one of π of π₯.

We have a graph of the function π of π₯ equals π₯ minus three, which we can use to find this limit graphically. The value of π₯ that weβre interested in is negative one, which we highlight on the π₯-axis. And we want to know what value π of π₯ approaches as π₯ approaches this value, negative one.

There are two directions in which π₯ can approach negative one. π₯ can approach negative one from below; that is, with π₯ always less than negative one but getting closer and closer to negative one. Or π₯ can approach negative one from above; that is, with π₯ always being greater than negative one but getting closer and closer to negative one.

We consider these two cases separately, so first the limit as π₯ tends to negative one from below of π of π₯. We pick a value of π₯ less than negative one, letβs say negative three, and we see that the point negative three, negative six lies on the graph. And so π of negative three is negative six.

Of course we could also have seen this from the definition of the function. But itβs more easy to see from the graph that as we choose other values of π₯ getting closer and closer to negative one, but always staying at less than negative one, that π of π₯ gets closer and closer to negative four. Therefore, the limit as π₯ tends to negative one from below of π of π₯ is negative four.

How about the limit as π₯ tends to negative one from above of π of π₯? Itβs the same story. This time we pick a value of π₯ is greater than negative one and see what happens when π₯ approaches negative one from the other direction; that is, with π₯ always being greater than negative one. We see that for this direction, also, the limit is negative four.

For a value of π₯ which is greater than negative one but very close to negative one, the value of π of π₯ is very close to negative four. If the limits from below and above have the same value, and in our case they do, we donβt need to specify whether weβre approaching the limits from below or above.

We can just say that the limit, period, is this value. And so the limit as π₯ tends to negative one, period, of π of π₯ is equal to negative four. Notice that in this case, the limit as π₯ tends to negative one of π of π₯ was just π evaluated at π₯ equals negative one. This isnβt true for all functions, but it is true for a large number of functions, including all linear functions whose graphs are straight lines.