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 (3 sin 4x) / 5x

Accepted Answer

Welcome back, everyone. Calculate the limit. Apply Lato's rule if necessary. Limit as X approaches 0 of 2 of 5 x divided by 3 X. We're given for answer choices A says 0, B 2/3, C 5/3, and D 103. So let's rewrite the given limit, limit as X approaches 0 of 2. Of 5 Xs divided by 3 acts. We always begin with direct substitution. If we substitute X equals 0, we're going to get 2 of 0 because 5 multiplied by 0 is 0, divided by 3 multiplied by 0. Sign of 0 is 0, so we're going to get 0 divided by 0, which is an indeterminate form. And Lapito's rule says that if we get an indeterminate form, Such as 0 divided by 0 or infinity divided by infinity, then we can apply the rule. So, according to the rule, we have to take the derivative of the top function and the derivative of the bottom function. So that our new limit is limit as X approaches 0. The derivative of 2, sin of 5 X. So what is the derivative of sine of 5 X? Well, we have to recall that the derivative of sine is cosine, so we get cosine of 5 x, and now we have an angle that is a linear function. So we have to apply the chain rule and multiply by the derivative of 5 X, which is 5. So we get two multiplied by cosine of 5x multiplied by 5 because the derivative of 5 X is 5, right? Now in the denominator, the derivative of X is 1, so 3 multiplied by 1 is 3, and our new limit becomes limit as x approaches 0 of 10, cosine of 5 x divided by 3. Now, let's apply the direct substitution once again to see if we got a finite limit. So that's 10 multiplied by cosine of 0. Divided by 3. Cosine of 0 is 1, so we get 10/3, which is indeed a finite number. So we have arrived at the final limit. So the correct answer choice, this problem corresponds to the answer choice D. 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 (3 sin 4x) / 5x

Key Concepts

Limits

L'Hôpital's Rule

Trigonometric Limits

Watch next