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Q.
- Identify Problem: We are asked to find the limit of the function as approaches . This is a limit problem involving trigonometric and square root functions.
- Recognize Indeterminate Form: First, we notice that direct substitution of into the function gives us a indeterminate form. This means we need to apply some algebraic manipulation or a limit law to evaluate the limit.
- Apply Algebraic Manipulation: To resolve the indeterminate form, we can multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of is .
- Multiply by Conjugate: Multiplying the numerator and denominator by the conjugate, we get:
- Simplify Denominator: Simplify the denominator using the difference of squares formula: . This gives us:
- Combine Like Terms: Simplify the denominator further:
- Distribute and Simplify Numerator: Combine like terms in the denominator:
- Evaluate Each Term: Now, we can simplify the numerator by distributing over the sum inside the square roots:
- Correct Error and Re-evaluate: We can now evaluate the limit of each term separately as approaches . For the first term, , as approaches , approaches and approaches . For the second term, , as approaches , approaches and approaches .
- Correct Error and Re-evaluate: We can now evaluate the limit of each term separately as approaches . For the first term, , as approaches , approaches and approaches . For the second term, , as approaches , approaches and approaches .However, we realize that there is a mistake in the previous step. The limit of as approaches is not , but rather . This means that the numerator approaches as approaches . We need to correct this error and re-evaluate the limit.
