Find the limits in Exercises 53–58. Write ∞ or −∞ where approp...

Question

Find the limits in Exercises 53–58. Write ∞ or −∞ where appropriate.

lim x/(x² − 1) as

d. x→−1⁻

Accepted Answer

Welcome back everyone. Find the limit as X approaches -2 from the left. We're given 4 answer choices A says -4, B 0, C negative infinity, and D infinity. And the limit that we want to evaluate is limit as X approaches -2 from the left of X divided by X2 minus 4. We can simply adapt to the direct substitution, assuming that our function is continuous at -2, which is going to give us -2 divided by -2 quad minus 4, which gives us 0 in the denominator, and division by 0 is undefined, right? So essentially what we want to do is understand that the limit is going to be infinity since we have a number divided by an infinitely small number approaching 0, right? So what we want to do is simply understand whether this is going to be positive or negative infinity, and the way to do that is simply by factoring out the denominator. So in the numerator we have X, and in the denominator we have a difference of squares. X2 minus 22, which in a factored form can be written as x minus 2 multiplied by X + 2. And now, if we carefully attempt the right substitution, we're going to get -2 in the numerator, and for the denominator, we have X minus 2, so -2 minus 2 gives us -4, multiplied by -2 from the left plus 2. Now what does that mean? If X approaches -2 from the left, this means that X is slightly less than -2, right? So if we add 2 to a number that is slightly less than -2, this means that we're going to get a negative number. So X + 2 is going to approach 0 from the left, right? And now from here we can take care of the signs. Notice that we have a negative number. Another negative number and the 3rd negative number. So, a negative number divided by the product between two negative numbers is basically a negative number divided by a positive number, which gives us a negative number. And that number is, or basically is represented by infinity, right, because whenever we have a number in the numerator and we're dividing by an infinitely small number that is close to 0, this means that the answer is infinity. So the whole idea of this problem was to determine the sign. Is it positive or negative infinity? And we have shown that it is negative infinity in this case, which corresponds to the answer of choice C. Thank you for watching.

Find the limits in Exercises 53–58. Write ∞ or −∞ where appropriate.

lim x/(x² − 1) as

d. x→−1⁻

Key Concepts

Limits

One-Sided Limits

Rational Functions

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