17–83. Limits Evaluate the following limits. Use l’Hôpital’s R...

Question

17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

lim_x→ 0⁺ (1 - ln x) / (1 + ln x)

Accepted Answer

Welcome back, everyone. Evaluate the following limit. Apply Lato's rule if necessary. Limit as X approaches 0 from the right of LNX minus 2 divided by LNX + 1. We given for answer choices A says -1, B 0, C1 and D infinity. So let's go ahead and rewrite the original limits. Limit as X approaches 0 from the right of LNX minus 2 divided by LNX + 1. We always begin with direct substitution. So essentially, what we can do is visualize the natural log function. Because we have the right side of limit, right? So this is really important. And specifically, we have an intersection point with the x axis, that would be 1.0. We are essentially approaching a.0 from the right and as we can see, our function tends towards negative infinity. So, In the numerator, we're going to get negative infinity minus 2, which is negative infinity, divided by a lot of 0 from the right. That's once again negative infinity plus 1, that's still negative infinity. And this is an indeterminate form, right? Essentially, if we have positive or negative infinity divided by positive or negative infinity, such an indeterminate form. Allows us to apply Lapido's rule, which says that we have to evaluate the derivative of the top function, so LNX minus 2 derivative and the derivative of the bottom function LNX + 1. So in this case, we are going to get a new limit. But its result will correspond to the limit of the actual function, right? So the derivative of LNX is 1 divided by X, the derivative of -2 is 0, and in the denominator we get another 1 divided by X, right? So 1 divided by X divided by 1 divided by X. Can be simplified as one, right, because we're dividing a number by itself. Well done. Now, this is a finite number, meaning we have successfully evaluated the limit and the correct answer to the problem corresponds to the answer choice C. Thank you for watching.

17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

lim_x→ 0⁺ (1 - ln x) / (1 + ln x)

Key Concepts

Limits

L'Hôpital's Rule

Natural Logarithm (ln)

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