Limits And Derivatives

Given,
On differentiating both sides w.r.t. , we get


Put , we get

[using L' Hospital's rule]

Step 1: Understanding the Concept:
This is a standard trigonometric limit that results in an indeterminate form . It can be solved using trigonometric identities or L'Hôpital's Rule.

Step 2: Key Formula or Approach:
Use the identity or apply L'Hôpital's Rule twice.

Step 3: Detailed Explanation:
1. Method 1: Trigonometric Identity Multiply the denominator by to match the angle: 2. Method 2: L'Hôpital's Rule Differentiate numerator and denominator: Differentiate again:

Step 4: Final Answer
The limit is .

Step 1: Understanding the Concept:
This limit involves the behavior of exponential growth as approaches infinity. In a sum of exponential terms, the term with the largest base dominates the growth of the entire expression.

Step 2: Detailed Explanation:
1. Let . 2. Factor out the largest base, which is : 3. Distribute the exponent : 4. As : - - - 5. The limit evaluates to:

Step 3: Final Answer
The limit is . (Note: If the limit was , the answer would be the geometric mean ).

Step 1: Direct substitution
Substitute : Thus, the limit is of the form (indeterminate).
Step 2: Apply L'Hôpital's Rule
Differentiate numerator and denominator:
Step 3: Evaluate the new limit
Substitute : Final Answer: