Use l’Hôpital’s Rule to find the limits in Exercises 85–108.

Question

88. lim(x→0) (tan x)/(x + sin(x))

Accepted Answer

Welcome back, everyone. Find the limit using Lapto's rule. Limit as x approaches 0 of tangent X divided by X + 2 X. A says 1/2, B 1/3, C 1/4, and D1. For this problem, let's go ahead and perform the direct substitution to check if we get the correct indeterminate form for Lapito's rules be applied. So we're going to rewrite our limit. We're going to attempt the direct substitution which gives us tangent of 0 and that's 0. Divided by 0 + 2 of 0, which also gives us 0. So we get an indeterminate form 0 divided by 0, which means that we can apply Lapito's rule. Lapito's rule says that we want to differentiate the numerator and the denominator. So, let's go ahead and find each derivative. In the numerator, we get the derivative of tangent, which is c 2 X. In the denominator, we get the derivative of X, which is 1 + 2 multiplied by the derivative of sine, which is cosine. So we get 1 plus 2 cosine X. And now we can again perform direct substitution. So we get a limit as x approaches 0. What is C2? Well, it is 1 divided by cosine squared, so we can write 1 divided by cosine squared of X, which is multiplied by 1 + 2 cosine x. And now let's go ahead and substitute X equals 0. We're going to get 1 in the numerator and in the denominator we get cosine of 0, which is 1. 1 squared multiplied by 1 + 2 multiplied by cosine of 0, so 2 multiplied by 1 gives us 2, right? And now performing the calculations, we get 1 divided by 1 which is multiplied by 3. This is 1/3. So the correct answer is option B. Thank you for watching.

Use l’Hôpital’s Rule to find the limits in Exercises 85–108.88. lim(x→0) (tan x)/(x + sin(x))

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Use l’Hôpital’s Rule to find the limits in Exercises 85–108.
88. lim(x→0) (tan x)/(x + sin(x))