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Evaluate the limit, if it exists.

$ \displaystyle \lim_{x \to 2}\frac{x^2 - 4x + 4}{x^4 - 3x^2 - 4} $

$\lim _{x \rightarrow 2} \frac{x^{2}+4 x+4}{x^{4}-3 x-4}=0$

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Okay, we want to find a limit of this function as X approaches to. Uh the temptation would be to substitute to directly in for X. But if you see what happens when you substitute to infer X in the denominator. Uh You're going to get it zero. Uh to to the 4th of 16 minus three. Times two squared would be 12, 16 minus 12 and then minus four. So 16 minus 12 minus four is zero. So if we directly substitute to infer ex uh into this expression trying to find a limit, it's not going to work because we're going to have zero into the in the denominator. Um So we can't directly substitute to wait for X. So what can we do? Um What we're gonna do is we're going to try to factor the expression in the numerator and the expression into the nominator. So we're gonna take the limit as X approaches to. Now let's look at the numerator. The numerator, X squared minus four, X plus four. That factors into X -2 times X -2. And then X to the fourth minus three, X squared minus four. Let's try to factor that. That should factor into X squared ah minus four times X squared plus one. Uh Let's check it real quick. Uh by using foil, X squared times X squared. Just texted the fourth. Uh Outer would be one X squared dinner would be negative four X squared. They combined to be negative three X squared last times last would be negative for. Okay so this works but we can factor this a little bit further. Let's write this one more time because we can factor the X squared minus four now X squared +11 factor. But x squared minus four. Wolf factor. So we're taking the limit as X approaches to Numerator was X -2 times X -2 denominator, X squared minus four is going to factor into x minus two Times X-plus two and we still have the times X squared plus one. All right, let's pause for a moment and uh look at what we've done so far. We want to find the limit of this expression as X approaches to. We saw that if we try to do it the easy way by directly substituting two and for x, you're going to get zero in the in the denominator. So that's not going to help us find the limit. So next we're trying the approach where we're factoring the numerator uh and the denominator and that's currently where we are x squared minus four, X plus four factors into x minus two times x minus two. The X to the fourth minus three, X squared minus four. Factored into x squared minus four times X squared plus one. And then we saw that we could uh factor the X squared minus four. A little bit further into x minus two times X plus tooth. So the denominator, X squared minus four times X squared plus one is really x minus two times X plus two times X squared plus one. So the limit of our function as X approaches to is going to equal the limit of this expression as X approaches to Now we can do a little simplifying. Uh we have an X -2 factor in the numerator and the denominator. That can be canceled. Uh we cannot cancel those if X -2 is zero. Okay, you can only cancel factors in the numerator and denominator as long as there are non zero. Well X minus two will be non zero, X minus two would only equal zero in excess to. We're taking the limit as X approaches to X does not get to equal to. Uh So we are allowed to cancel out uh these x minus two factors. Uh So what does that leave us? That leaves us taking the limit as X approaches to of x minus two over X plus two times X squared plus one. Now we can directly substitute to And for X, what happens to the numerator As X approaches to? As X approaches to X -2 approaches to -2 which is zero. What happens to this? X plus two factor in the denominator. As X approaches to as X approaches to X plus two approaches two plus two, which is four. How about the X squared plus one factor as X approaches to X squared plus one approaches two squared plus one, which is five. And so this expression as X approaches to this expression will approach 0/4 times five, 0/20 Which of course is zero. And so the limit of this function as X approaches to equals zero.