Answer :

Final answer:

The limit of the function √(x+1)-2/x-3 as x approaches 3 is 1/4 after applying L'Hopital's Rule to convert indeterminant form to determinate form.

Explanation:

To find the limit of the function √(x+1)-2/x-3 as x approaches 3, it's important to apply the rules of limits and calculus. Since the function seems to be undefined at x=3, we're essentially looking for a hole or removable discontinuity at that point.

However, if you substitute x=3 directly into the numerator, you get the indeterminant form 0/0. This situation calls for the application of L'Hopital's Rule which states that if the limit of a function leads to indeterminant form (0/0 or ∞/∞), then the limit of that function is equal to the limit of the derivative of the numerator divided by the derivative of the denominator.

This results in the derivative of the numerator (√(x+1)-2) being (1/2)(x+1)^-1/2 and the derivative of the denominator (x-3) being 1. Then the limit as x approaches 3 of the fraction of these derivatives is 1/4, choice C.

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