Limits And Derivatives

Concept: Calculus - Limits at Infinity and Rationalization.

Step 1: Rationalize the main expression. The given limit is of the form . To resolve this, multiply and divide the expression within the brackets by its conjugate: . The numerator becomes . The expression is now: .

Step 2: Rationalize the new numerator. The numerator still leads to an indeterminate form. Rationalize it again by multiplying the numerator and denominator by its conjugate, . The new numerator becomes . The limit expression is now: .

Step 3: Factor out the highest powers of from the denominator. Factor out from the first bracket and from the second bracket to match the in the numerator. First bracket: extract to get . Second bracket: extract to get . Multiply these extracted factors: .

Step 4: Cancel and prepare to evaluate the limit. Substitute the factored denominator back into the limit: . The terms cancel out completely, leaving: .

Step 5: Evaluate the limit as . As approaches infinity, approaches 0. Substitute 0 for all terms: . $ $

Given Limit:
We are tasked with solving the following limit:

Step 1: Express Trigonometric Functions in Terms of Sine and Cosine:
Recall that: So, we can rewrite the limit as:

Step 2: Simplify the Expression:
Simplify the denominator and the trigonometric expressions:

Step 3: Apply Small Angle Approximations:
As , we can use the approximations and . Using these approximations: Substituting these into the expression:

Final Answer:
Therefore, the value of the limit is: