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Q.
- Identify Limit Form: Identify the form of the limit to determine the method of solution.As approaches , both and approach . Therefore, and both approach , which is . This gives us a indeterminate form, which suggests that we can use L'Hôpital's Rule to evaluate the limit.
- Apply L'Hôpital's Rule: Apply L'Hôpital's Rule. L'Hôpital's Rule states that if the limit of as approaches a value is in the indeterminate form or , then the limit is the same as the limit of as approaches , provided that the derivatives exist and the limit of is determinate.
- Differentiate Numerator and Denominator: Differentiate the numerator and denominator separately.The derivative of with respect to is , using the chain rule.The derivative of with respect to is , also using the chain rule.
- Simplify Derivatives: Simplify the derivatives.The derivative of simplifies to .The derivative of simplifies to .
- Apply L'Hôpital's Rule: Apply L'Hôpital's Rule by taking the limit of the derivatives.
- Evaluate Simplified Expression: Evaluate the limit of the simplified expression.As approaches , approaches and approaches because is approximately equal to for small values of .Therefore, the limit becomes .
- Simplify and Evaluate Limit: Simplify the expression and evaluate the limit.The 's cancel out, and we are left with . Since there are no terms left, the limit is simply .
