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Given that the following graph represents the function 𝑓(𝑥) = (𝑥² − 4)/(𝑥 + 2), determine lim_(𝑥 → −2) 𝑓(𝑥).
Given that the following graph represents the function 𝑓 of 𝑥 equals 𝑥 squared minus four over 𝑥 plus two, determine the limit of 𝑓 of 𝑥 as 𝑥 approaches negative two.
Let’s take a look at our graph. And we can see that it looks very much like the graph of a straight line. And in fact, it is, with one small difference, this opened dot here with coordinates negative two, negative four. And this opened dot represents a hole in the graph or a gap in the straight line. So 𝑓 of negative two is not negative four.
There’s a gap in the graph where we’d like to read off from. And because there’s no solid start on the graph with 𝑥-coordinates negative two, in fact, 𝑓 of negative two is undefined. We’ve used the graph of 𝑓 of 𝑥 to determine this. But we could also use the algebraic rule that defines 𝑓 of 𝑥 in the question.
𝑓 of 𝑥 equals 𝑥 squared minus four over 𝑥 plus two. We can plug negative two into this rule. And simplifying this expression on the right-hand side, we get zero over zero. Here we’ve used the fact that negative two squared is four. And so the numerator four minus four is zero. And similarly, the denominator negative two plus two is just zero.
Zero over zero is an indeterminate form, which is undefined. And so our algebraic rule agrees with our graph. 𝑓 of negative two is undefined. How does this affect the value of the limit of 𝑓 of 𝑥 as 𝑥 approaches negative two?
The value of 𝑓 of negative two doesn’t affect this limit at all. So 𝑓 of negative two could be negative four or three or a billion or as it is here, undefined. And it doesn’t affect the limit as 𝑥 tends to negative two of 𝑓 of 𝑥. What is important are the values of 𝑓 of 𝑥 when 𝑥 is very near negative two.
The limit of 𝑓 of 𝑥 as 𝑥 approaches negative two is the value that 𝑓 of 𝑥 gets closer and closer to as 𝑥 approaches negative two. We can see from the graph that 𝑓 of negative three is negative five. And as 𝑥 gets closer to negative two, 𝑓 of 𝑥 gets closer and closer to negative four.
The same is true if we approach 𝑥 equals negative two from the right. 𝑓 of 𝑥 gets closer and closer to negative four. And so this is the value of our limit. The limit of 𝑓 of 𝑥 as 𝑥 approaches negative two is negative four.
By choosing a value of 𝑥 close enough to negative two, you can make 𝑓 of 𝑥 as close as you want to negative four. The fact that the function 𝑓 is actually undefined in negative two is not important.
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