Suppose f(x)→100 and g(x)→0, with g(x)

Question

Suppose f(x)→100 and g(x)→0, with g(x)<0 as x→2. Determine lim x→2 f(x) / g(x).

Accepted Answer

Welcome back, everyone. Given that MX approaches -75, as X approaches 5, and NX approaches 0 with N of X being negative, as X approaches 5, what is the limit of M X divided by N of X as X approaches 5 and Given4 answer choices A says negative infinity, B 75, C infinity, and D 0. So let's define the limit itself. We want to evaluate the limit as x approaches 5 of M X divided by N of X. And according to the context of the problem. We know that M of X approaches -75 S X approaches 5, so we can say that the limit as X approaches 5. Of M of X is equal to -75. And we also can say that. The limit is x approaches 5 of N of X. Is going to be equal to 0. And specifically, we can say that this number is going to be 0 negative, right? 0 from the left because N of X is less than 0 as X approaches 5. So now what we're going to do is basically apply the properties of limits and we're going to rewrite the original limit as a ratio of 2 limits. In the numerator we're going to take the limit as x approaches 5 of M X, and in the denominator we're going to get limit as x approaches 5 of N of X. And now if we use the direct substitution. We're going to get -75 in the numerator and in the denominator we get 0 from the left. So now whenever we divide a number by 0, when we are considering limits, well, we get a limit that corresponds to infinity. But the most important part is to understand whether we should include a positive or a negative time. Now notice that we are dividing a negative number by 0. We are approaching from the left, meaning also a negative number. It is simply a very small negative number. So whenever we divide a negative number by another negative number, we're going to get a positive number, and that number is going to be infinitely large because we are dividing. 75 by 0, right? Therefore, we can conclude that the correct answer to this problem corresponds to the answer choice C. Thank you for watching.

Suppose f(x)→100 and g(x)→0, with g(x)<0 as x→2. Determine lim x→2 f(x) / g(x).

Key Concepts

Limits

Indeterminate Forms

L'Hôpital's Rule

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