Limits Of Trigonometric Functions

Step 1: Define substitution
Let

x%20%3D%20%5Ctheta%20-%20%5Cfrac%7B%5Cpi%7D%7B2%7D

, then as

%5Ctheta%20%5Cto%20%5Cfrac%7B%5Cpi%7D%7B2%7D

, we use the approximation:

%5Ctan%5Ctheta%20%5Capprox%20%5Cfrac%7B1%7D%7Bx%7D%20%5Cquad%20%5Ctext%7Bas%20%7D%20x%20%5Cto%200.

Step 2: Substitute and Simplify
Substituting in the given expression:

8%5Ctan%5E4%5Ctheta%20%2B%204%5Ctan%5E2%5Ctheta%20%2B%205%20%3D%208%5Cleft(%5Cfrac%7B1%7D%7Bx%7D%5Cright)%5E4%20%2B%204%5Cleft(%5Cfrac%7B1%7D%7Bx%7D%5Cright)%5E2%20%2B%205

%3D%20%5Cfrac%7B8%7D%7Bx%5E4%7D%20%2B%20%5Cfrac%7B4%7D%7Bx%5E2%7D%20%2B%205.

For the denominator:

(3%20-%202%5Ctan%5Ctheta)%5E4%20%3D%20(3%20-%202%5Ccdot%5Cfrac%7B1%7D%7Bx%7D)%5E4%20%3D%20%5Cleft(%5Cfrac%7B3x%20-%202%7D%7Bx%7D%5Cright)%5E4.

Step 3: Compute the limit
Taking the limit, the dominant terms give:

%5Clim_%7Bx%20%5Cto%200%7D%20%5Cfrac%7B%5Cfrac%7B8%7D%7Bx%5E4%7D%20%2B%20%5Cfrac%7B4%7D%7Bx%5E2%7D%20%2B%205%7D%7B%5Cleft(%5Cfrac%7B3x%20-%202%7D%7Bx%7D%5Cright)%5E4%7D%20%3D%20%5Cfrac%7B8%7D%7B16%7D%20%3D%20%5Cfrac%7B1%7D%7B2%7D.

Thus, the correct answer is

%5Cboxed%7B%5Cfrac%7B1%7D%7B2%7D%7D

Chapter Questions
1 MCQs