Question

Use L'Hôpital's rule to determine the following limit, where 'a' is regarded as a constant:

a) Limit as x approaches a of (eˣ - eᵃ)/(x - a)
b) Limit as x approaches a of (ln(x) - ln(a))/(x - a)
c) Limit as x approaches a of (sin(x - a))/(x - a)
d) Limit as x approaches a of (1 - cos(x - a))/(x - a)

Answer 1

Using L'Hôpital's rule, we find the limits of the given functions by differentiating the numerator and denominator. The limits are e^a, 1/a, 1, and 0, respectively.

Step-by-step explanation:

Applying L'Hôpital's rule is a method used to find limits of indeterminate forms, such as 0/0 or ∞/∞. When we apply L'Hôpital's rule to the given limit problems, we take the derivative of the numerator and denominator separately and then compute the limit again.

1. For the limit as x approaches a of (ex - ea)/(x - a), both the numerator and denominator approach 0 as x approaches a. By taking the derivative of the numerator and denominator, we get the limit is equal to ea.
2. The limit as x approaches a of (ln(x) - ln(a))/(x - a) also results in an indeterminate form. Applying L'Hôpital's rule, we take the derivative of the numerator and denominator to find the limit is 1/a.
3. In the limit as x approaches a of (sin(x - a))/(x - a), the limit is directly known without needing L'Hôpital's rule as it is a well-known trigonometric limit which equals 1.
4. Lastly, for the limit as x approaches a of (1 - cos(x - a))/(x - a), we can apply L'Hôpital's rule to determine that the limit is 0.