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.

49. lim (x → 0) (x - sin x) / (x tan x)

Accepted Answer

Welcome back, everyone. Evaluate the limit using Lapital's rule. Limit as X approaches 0 of E to the power of X minus X minus 1 divided by X2. For this problem, let's rewrite the limit. And let's remember that whenever we want to apply Lapto's rule, we want to first we'll substitute X equals 0 for every X. And we're going to get e to the power of 0 minus 0 minus 1, which is 1 minus 0 minus 1, and that's 0. So we get a 0 in the numerator. And 0 squared or basically 0 in the denominator. 0 divided by 0 is a suitable form for Laptose rule to be applied. And Lapito's rule says that we want to take the derivative of the numerator. So the derivative of each, the power of X minus X minus 1. And divide by the derivative of the denominator, so the derivative of X squared. Let's find each derivative. We get limit as X approaches 0 of E to the power of X. -1, 0. So basically eats the power of X minus 1 is the derivative of the numerator. And for the denominator, the derivative of X squared is to x using the power rule. Now, substitute X equals 0 for every X again we get E to the power of 0 minus 1. Which is 1 minus 1 or 0 in the numerator, and 2 multiplied by 0 gives us 0 in the denominator. So we still get the same in determinant form, which means Applying Lapito's rule once again. We want to differentiate E to the power of X minus 1, and divide by the derivative of 2 X. So now we're going to get limit as X approaches 0 of E to the power of X minus 0, which is E to the power of X. Divided by 2 because the derivative of X is 1. Once again, we're going to attempt the direct substitution. It's the power of 0. Gives us 1 in the numerator and we have 2 in the denominator. And now this is a finite value, meaning the final limit is 12. That's our finally answering and thank you for watching.

Use l’Hôpital’s rule to find the limits in Exercises 7–52.49. lim (x → 0) (x - sin x) / (x tan x)

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Use l’Hôpital’s rule to find the limits in Exercises 7–52.
49. lim (x → 0) (x - sin x) / (x tan x)