To find the limit of the given function [tex]\(\frac{x^2 - \sqrt{x}}{\sqrt{x} - 1}\)[/tex] as [tex]\( x \)[/tex] approaches 1, we can follow these steps:
1. Substitute [tex]\(x = 1\)[/tex] to Check Direct Evaluation:
Directly substituting [tex]\(x = 1\)[/tex] into the function:
[tex]\[
\frac{1^2 - \sqrt{1}}{\sqrt{1} - 1} = \frac{1 - 1}{1 - 1}
\][/tex]
yields an indeterminate form [tex]\(\frac{0}{0}\)[/tex].
2. Simplify the Expression:
To resolve the indeterminate form, we need to simplify the given expression:
[tex]\[
\frac{x^2 - \sqrt{x}}{\sqrt{x} - 1}
\][/tex]
3. Factorization and Simplification:
Notice that the numerator [tex]\(x^2 - \sqrt{x}\)[/tex] can be written in a different form. First, let us express [tex]\(\sqrt{x}\)[/tex] as [tex]\(y\)[/tex], such that [tex]\(y = \sqrt{x}\)[/tex]. Then [tex]\(x = y^2\)[/tex]. We rewrite the function in terms of [tex]\(y\)[/tex]:
[tex]\[
\frac{(y^2)^2 - y}{y - 1} = \frac{y^4 - y}{y - 1}
\][/tex]
Now, factor the numerator:
[tex]\[
y^4 - y = y(y^3 - 1) = y(y - 1)(y^2 + y + 1)
\][/tex]
We substitute back [tex]\(y = \sqrt{x}\)[/tex] to get:
[tex]\[
\frac{\sqrt{x} \left( (\sqrt{x})^3 - 1 \right)}{\sqrt{x} - 1} = \frac{\sqrt{x} (\sqrt{x} - 1) (\sqrt{x}^2 + \sqrt{x} + 1)}{\sqrt{x} - 1}
\][/tex]
4. Cancel Common Factors:
Since [tex]\(\sqrt{x} - 1\)[/tex] is a common factor in the numerator and the denominator, we can cancel it out:
[tex]\[
\frac{\sqrt{x} (\sqrt{x}^2 + \sqrt{x} + 1)}{(\sqrt{x} - 1)}
\][/tex]
After canceling [tex]\(\sqrt{x} - 1\)[/tex], we get:
[tex]\[
\sqrt{x} (\sqrt{x}^2 + \sqrt{x} + 1)
\][/tex]
5. Evaluate the Simplified Expression at [tex]\(x = 1\)[/tex]:
Now, substitute [tex]\(x = 1\)[/tex] into the simplified expression:
[tex]\[
\sqrt{1} \left( (\sqrt{1})^2 + \sqrt{1} + 1 \right)
\][/tex]
or equivalently:
[tex]\[
1 \left( 1^2 + 1 + 1 \right) = 1 \left( 1 + 1 + 1 \right) = 1 \times 3 = 3
\][/tex]
Thus, the limit is:
[tex]\[
\lim_{x \rightarrow 1} \frac{x^2 - \sqrt{x}}{\sqrt{x} - 1} = 3
\][/tex]
