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

Question

91. lim(x→π/2⁻) (sec(7x))(cos(3x))

Accepted Answer

Welcome back, everyone. Evaluate the limit using Lapito's rule. Limit as X approaches divided by 2 from the left of C 5 X cosine 3X. A says 3/5, B negative 5/3, C 35, and D 5/3. Let's begin by writing the limit. We have a limit as x approaches pi divided by 2 from the left of C count of 5 X multiplied by cosine of 3 X. Now, what we can do is simply rewrite sequent in terms of cosine, right? So we're going to write the limit as X approaches by divided by 2 from the left of cosine of 3 X, and sequent of 5 X is 1 divided by cosine of 5 X, right? Because we're multiplying by 1 divided by cosine of 5 X we can simply divide by cosine of 5 X. If we substitute X equals pi divided by 2 for every X, we're going to get cosine of 3 pi divided by 2, which is 0 in the numerator, and in the denominator we're going to get cosine of 5 pi divided by 2, which also gives us 0. So we get an interterminant form 0 divided by 0, which means that lapital stroll can be applied immediately. According to Lapto's rule, we want to differentiate the numerator and the denominator. The derivative of cosine is negative sign, so we're going to get negative sign of 3 X, and according to the chain rule, we're multiplying by the derivative of 3 X, which is 3, right? In the denominator We want to differentiate cosine of 5 X. So similarly, we get negative sign of 5 X multiplied by 5. And now what we're going to do is cancel out the two negative signs. We can also factor out 3/5. And we're going to attempt direct substitution once again. We're going to get a sign of 3 pi divided by 2, divided by. Sign of 5 pi divided by 2. So now we get 3/5 multiplied by sin of 3 pi divided by 2 is going to be equal to -1. And sine of 5 pi divided by 2 is going to be 1. Notice that now we're getting a finite value which is 3/5 multiplied by -1, and this is equal to -35. So the correct answer for this problem corresponds to the answer choice C. Thank you for watching.

Use l’Hôpital’s Rule to find the limits in Exercises 85–108.91. lim(x→π/2⁻) (sec(7x))(cos(3x))

Watch next

Use l’Hôpital’s Rule to find the limits in Exercises 85–108.
91. lim(x→π/2⁻) (sec(7x))(cos(3x))