Question

Evaluate: lim h->0 (√(x+h) - √x) / h

Answer 1

Rationalizing the numerator, it is found the result of the limit is given as follows:

$\lim_{h \rightarrow 0} \frac{(\sqrt{x + h} - \sqrt{x})}{h} = \frac{1}{2\sqrt{x}}$

What is a limit?

A limit is given by the value of function f(x) as x tends to a value.

The first step to find the limit is replace the variable by the value, hence:

$\lim_{h \rightarrow 0} \frac{(\sqrt{x + h} - \sqrt{x})}{h} = \frac{0}{0}$

Undefined limit, hence we have to find another way. The numerator includes square roots, hence it should be rationalized using the subtraction of perfect squares, as follows:

$\lim_{h \rightarrow 0} \frac{(\sqrt{x + h} - \sqrt{x})}{h} \times \frac{(\sqrt{x + h} + \sqrt{x})}{(\sqrt{x + h} + \sqrt{x})} = \lim_{h \rightarrow 0} \frac{(\sqrt{x + h})^2 - (\sqrt{x})^2}{h(\sqrt{x + h} + \sqrt{x})} = \lim_{h \rightarrow 0} \frac{x + h - x}{h(\sqrt{x + h} + \sqrt{x})} = \lim_{h \rightarrow 0} \frac{h}{h(\sqrt{x + h} + \sqrt{x})}$

Then we simplify the h, so:

$\lim_{h \rightarrow 0} \frac{1}{\sqrt{x + h} + \sqrt{x}} = \frac{1}{2\sqrt{x}}$

More can be learned about limits at brainly.com/question/26270080

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