Use l’Hôpital’s rule to find the limits in Exercises 7–52.
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Question

Use l’Hôpital’s rule to find the limits in Exercises 7–52.

30. lim (θ → 0) ((1/2)^θ - 1) / θ

Accepted Answer

Welcome back, everyone. Evaluate the limit. Limit as X approaches 0 of 3 to the power of X minus 1 divided by X. A says 0. BLN of 3. C 1 divided by LN of 3, and D the limit does not exist. Let's begin by rewriting the limit, the limit as x approaches 0 of tweets the power of X minus 1 divided by X. And let's remember that the first step is to assume that our function is continuous at that point. So we can substitute X equals 0, and we get tweets the power of 0 minus 1 divided by 0. And that's basically 0 divided by 0. It is an indeterminate form. And this in the terminal form is basically saying to us that we can apply Lapino's rule. According to Lapto's rule, we essentially want to differentiate the numerators, so we want to take the derivative of 3 to the power of X minus 1 and divide by the derivative of the denominator. Let's go ahead and find each derivative. We're going to get limit as X approaches 0 off. The derivative of tweets the power of X is tweets the power of XLN 3. The derivative of -1 is 0. And in the denominator, the derivative of X is simply one, right? So we end up with the limit as X approaches 0 of tweets the power of X and N of 3. And now we repeat the process we. Essentially try to substitute X equals 0, and we are going to get 3 to the power of 0 multiplied by LN of 3. And now it is finite, right? We get one multiplied by LN of 3, which is LN of 3. So the correct answer for this problem is B. Thank you for watching.

Use l’Hôpital’s rule to find the limits in Exercises 7–52.30. lim (θ → 0) ((1/2)^θ - 1) / θ

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Use l’Hôpital’s rule to find the limits in Exercises 7–52.
30. lim (θ → 0) ((1/2)^θ - 1) / θ