Infinite Limits

Find the lim...

Question

Infinite Limits

Find the limits in Exercises 37–48. Write ∞ or −∞ where appropriate.

lim x→0 (−1) / (x² (x + 1))

Accepted Answer

Welcome back, everyone. Find the limit as x approaches 0 for the function F of X equals 3 divided by x2 multiplied by x + 2. We're given 4 answer choices A says infinity, B 0 C negative infinity, and D does not exist. So let's write down the limit limit as x approaches 0 of 3 divided by x2 multiplied by x + 2. We're going to begin with the rank substitution, assuming that our function is continuous at X equals 0, right? So we're getting 3 divided by 0 squared multiplied by 0 + 2. Overall, we can say that we're dividing 3 by 0. If this is the case, whenever we divide a non-zero number by 0 in our attempted evaluation, we can say that our answer is going to be some sort of infinity. It might also not exist, but we will focus on this concept a bit later, right? But overall, we have to understand that a number divided by 0 in the attempted evaluation most likely leads to infinity. So now our goal is to understand whether it's positive or negative infinity or perhaps the limit does not exist. So what we want to do is simply analyze the limit from the left and from the right using the limit existence theorem. So we want to valueate limit as x approaches 0 from the left of F of X. What does that mean? Well, we can attempt 3. And divide that by 0 from the left squared multiplied by 0 + 2, which gives us 2, right? We already know that we divided by 0 is infinity we simply want to determine the sign. Now 0 from the left here squared. We are going to focus on this term, so if X approaches 0 from the left, it basically means that X is less than 0, right? It is slightly less than 0. But if we square a negative number, it is going to be positive, right? So we can say that. This expression becomes 3 divided by 0 from the right. A positive number, right? Because when we square a negative number, it becomes positive, multiplied by 2. Well, it's still a positive number. So a positive number divided by a positive number is going to give us a positive number, meaning the limit is positive infinity. And what about the limit as X approaches 0 from the right of F of X? Well, if we attempt the calculation, this gives us 3 divided by 0 from the right squared, which is just 0 from the right, multiplied by 2. So this also gives us a ratio of positive numbers, which is positive. So again, a positive infinity, because the limit from the left and from the right of zero is the same positive infinity, the limit indeed exists and is equal to positive infinity, the common limit of those two, meaning the correct answer to this problem corresponds to the answer choice A infinity. Thank you for watching.

Infinite Limits

Find the limits in Exercises 37–48. Write ∞ or −∞ where appropriate.

lim x→0 (−1) / (x² (x + 1))

Key Concepts

Limits

Infinite Limits

Behavior of Rational Functions

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