Question

Solve without lhopital lim x>pi sinx/x-pi

A) Apply Taylor series expansion.
B) Use numerical methods for integration.
C) Simplify using trigonometric identities.
D) Compute eigenvalues and eigenvectors.

Answer 1

To solve the limit lim x > π sinx/x - π without using L'Hospital's Rule, we can simplify using trigonometric identities.

Step-by-step explanation:

To solve the limit lim x > π sinx/x - π without using L'Hospital's Rule, we can simplify using trigonometric identities.

Solution:

1. Start by factoring out π from the denominator: sinx/x - π = (sinx - πx)/x.
2. Next, use the identity sin(A - B) = sin A cos B - cos A sin B to rewrite the numerator: sinx - πx = sin(x - π) = -sin(x).
3. Now, substitute this back into the expression: (sinx - πx)/x = -sin(x)/x.
4. Finally, take the limit as x approaches π: lim x > π -sin(x)/x.

By applying trigonometric identities, we have simplified the expression. The limit can now be evaluated using properties of limits or numerical methods.