To determine the limit as [tex]\( h \)[/tex] approaches 0 of the expression [tex]\(\frac{\sqrt{x+h} - \sqrt{x}}{h}\)[/tex], let's go through the steps of the calculation.
1. Understand the Expression: The given expression is [tex]\(\frac{\sqrt{x+h} - \sqrt{x}}{h}\)[/tex]. We need to find its limit as [tex]\( h \)[/tex] approaches 0.
2. Expression Analysis: As [tex]\( h \)[/tex] approaches 0, both [tex]\(\sqrt{x+h}\)[/tex] and [tex]\(\sqrt{x}\)[/tex] are finite and close to each other, so the numerator (the difference) approaches 0, while [tex]\( h \)[/tex] in the denominator also approaches 0. This hints at an indeterminate form [tex]\(\frac{0}{0}\)[/tex].
3. Apply Rationalization: To resolve this, we'll rationalize the numerator by multiplying and dividing by the conjugate of the numerator: [tex]\(\sqrt{x+h} + \sqrt{x}\)[/tex].
Consider:
[tex]\[
\lim_{h \to 0} \frac{\sqrt{x+h} - \sqrt{x}}{h} \times \frac{\sqrt{x+h} + \sqrt{x}}{\sqrt{x+h} + \sqrt{x}}
\][/tex]
which simplifies to:
[tex]\[
\lim_{h \to 0} \frac{(\sqrt{x+h} - \sqrt{x})(\sqrt{x+h} + \sqrt{x})}{h (\sqrt{x+h} + \sqrt{x})}
\][/tex]
4. Simplify the Numerator: Using the difference of squares, we get:
[tex]\[
(\sqrt{x+h})^2 - (\sqrt{x})^2 = (x+h) - x = h
\][/tex]
So the expression becomes:
[tex]\[
\lim_{h \to 0} \frac{h}{h(\sqrt{x+h} + \sqrt{x})}
\][/tex]
5. Cancel Common Terms: Cancel the [tex]\( h \)[/tex] term in the numerator and denominator:
[tex]\[
\lim_{h \to 0} \frac{1}{\sqrt{x+h} + \sqrt{x}}
\][/tex]
6. Evaluate the Limit: As [tex]\( h \)[/tex] approaches 0, [tex]\(\sqrt{x+h}\)[/tex] approaches [tex]\(\sqrt{x}\)[/tex]. Hence:
[tex]\[
\lim_{h \to 0} \frac{1}{\sqrt{x+h} + \sqrt{x}} = \frac{1}{\sqrt{x} + \sqrt{x}} = \frac{1}{2\sqrt{x}}
\][/tex]
Thus, the limit is:
[tex]\[
\lim_{h \to 0} \frac{\sqrt{x+h} - \sqrt{x}}{h} = \frac{1}{2\sqrt{x}}
\][/tex]
So, the final answer is [tex]\(\boxed{\frac{1}{2\sqrt{x}}}\)[/tex].
