Analyze the following limits and find the vertical asymptotes ...

Question

Analyze the following limits and find the vertical asymptotes of f(x) = (x − 5) / (x² − 25).

lim x → -5^- f(x)

Accepted Answer

Welcome back, everyone. Consider the function F of X, which is equal to X minus 11 divided by X2 minus 121. Determine the limit as x approaches -11 from the left of F of X and identify the vertical asymptotes of the function F. Let's begin with the limit. We want to identify the limit as x approaches -11 from the left, and we want to substitute the function. So that's x minus 11 divided by X2 minus 121. If we just substitute -11 for every X, we're going to get an indeterminate form, 0 divided by 0, right? And because this is an indeterminate form, well, we want to manipulate the function and rewrite it in a factored form. So that we can eliminate a possible hole in the function. So let's rewrite the limit, but now what we're going to do is simply factor out the denominator because it is a difference of squares. Let's understand that X2 minus 121 is essentially. X2 minus 11 squad and now applying the difference of squares formula we get a factored form x minus 11 for our first factor multiplied by x + 11 our second factor, and here we can understand that x minus 11 is the common factor that can be canceled out. So the limit becomes a limit as X approaches -11 from the left of 1 divided by X + 11, right? Now let's attempt the direct substitution once again in the numerator we have 1 in the denominator we're going to get -11 from the left plus 11, which is essentially 1 divided by 0. This is some sort of infinity because we're dividing a number by 0, and this gives us a limit of infinity. But now we need to understand if this infinity has a positive or a negative sign. So now, if we are approaching -11 from the left, this means that X is less than -11, and as a result, -11 from the left plus 11. is going to be less than 0, right? Essentially because we're adding 11 to a number that is less than -11. So this means that we're dividing a positive number by 0 from the left. We can also rewrite it as 1 divided by 0 from the left. A ratio of a positive and a negative number gives us a negative number. So the first answer is negative infinity. We got our limit. And now let's identify the vertical asymptotes. Well, we can take our function in its factored form and we can say that F of X. is equal to 1 divided by x + 11, where x is not equal to 11 because we have canceled out that factor, right? So essentially what we want to do is understand that vertical asymptotes make our function undefined, so we're setting our denominator to 0, X plus 11 equals 0, and this gives us X equals -11. Since this value doesn't make our, our numerator equal to zero as well, this is not a hole. This is indeed a vertical asymptote, right? Essentially, this is the vertical asymptote and now considering the other factor. X minus 11, we already know that we have canceled it out, right? So if we set it to 0, we're going to get X equals 11, and this is a hole because it also makes the numerator equal to 0. So this is a hole. But the previous one X equals -11 is a vertical asymptote, right? The reason why it's a vertical asymptote is because it makes the denominator equal to 0, the numerator is not equal to 0. But if we get a 0 divided by a 0 form, well, this gives us a whole. So what can we conclude about this problem? Well, we have two answers. The first one of them is the limit value. So limit as x approaches -11 from the left of F of X is equal to negative infinity. And now for the vertical asymptotes we can say that the only vertical asymptote has a form of x equals -11, and those are the answers to this problem. Thank you for watching.

Analyze the following limits and find the vertical asymptotes of f(x) = (x − 5) / (x² − 25).
lim x → -5^- f(x)

Key Concepts

Limits

Vertical Asymptotes

Factoring Polynomials

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