lim _(xarrow 3)(sqrt (2x+3)-x)/(3-x)

Question

Answer 1

To evaluate the limit

, we can use algebraic manipulation and L'Hôpital's Rule.First, let's rewrite the expression:

We notice that direct substitution of

results in an indeterminate form

. Therefore, we can apply L'Hôpital's Rule, which states that if \(\lim_{x \to c} \frac{f(x)}{g(x)}\) results in

or

, then:

First, we need to find the derivatives of the numerator and the denominator.For the numerator

:

Using the chain rule:

Thus,

For the denominator

:

Now, applying L'Hôpital's Rule:

Simplify the expression inside the limit:

Substitute

:

Therefore, the limit is:

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