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Find .Choose answer:(A) (B) (C) (D) The limit doesn't exist
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Q. Find .Choose answer:(A) (B) (C) (D) The limit doesn't exist
- Evaluate Limit Directly: First, we need to evaluate the limit directly by substituting into the expression to see if it results in an indeterminate form.\lim_{x \to 1}\frac{\sqrt{5x+4}-3}{x-1} = \frac{\sqrt{5(1)+4}-3}{1-1}\(\newline= \frac{\sqrt{9}-3}{0}= \frac{3-3}{0}= \frac{0}{0}\)This is an indeterminate form, so we cannot evaluate the limit by direct substitution.
- Resolve Indeterminate Form: To resolve the indeterminate form, we can use algebraic manipulation. Specifically, we can multiply the numerator and the denominator by the conjugate of the numerator to eliminate the square root.The conjugate of the numerator is .We multiply the original expression by .
- Simplify Using Conjugate: Now, we simplify the expression by multiplying out the numerators and denominators.= =
- Factor Out Numerator: We can now simplify the expression further by factoring out a from the numerator.
- Cancel Common Factor: Since is a common factor in the numerator and the denominator, we can cancel it out.
- Evaluate Simplified Expression: Now we can evaluate the limit by substituting into the simplified expression.= = =
- Final Answer: The final answer is (B) , which is the limit of the given expression as approaches .
