Calculate: \lim_{x arrow 1} ((\sqrt[3]{x^2}-2\sqrt[3]{x}+1)/((x-1)^2))

Expression: $\lim_{x \rightarrow 1} \left(\frac{ \sqrt[3]{{x}^{2}}-2\sqrt[3]{x}+1 }{ {\left( x-1 \right)}^{2} }\right)$

Evaluate the limits of numerator and denominator separately

$\begin{array} { l }\lim_{x \rightarrow 1} \left(\sqrt[3]{{x}^{2}}-2\sqrt[3]{x}+1\right),\\\lim_{x \rightarrow 1} \left({\left( x-1 \right)}^{2}\right)\end{array}$

Evaluate the limit

$\begin{array} { l }0,\\\lim_{x \rightarrow 1} \left({\left( x-1 \right)}^{2}\right)\end{array}$

Evaluate the limit

$\begin{array} { l }0,\\0\end{array}$

Since the expression $\frac{ 0 }{ 0 }$ is an indeterminate form, try transforming the expression

$\lim_{x \rightarrow 1} \left(\frac{ \sqrt[3]{{x}^{2}}-2\sqrt[3]{x}+1 }{ {\left( x-1 \right)}^{2} }\right)$

Use ${a}^{2}-2ab+{b}^{2}={\left( a-b \right)}^{2}$ to factor the expression

$\lim_{x \rightarrow 1} \left(\frac{ {\left( \sqrt[3]{x}-1 \right)}^{2} }{ {\left( x-1 \right)}^{2} }\right)$

Use ${a}^{3}-{b}^{3}=\left( a-b \right)\left( {a}^{2}+ab+{b}^{2} \right)$ to factor the expression

$\lim_{x \rightarrow 1} \left(\frac{ {\left( \sqrt[3]{x}-1 \right)}^{2} }{ {\left( \left( \sqrt[3]{x}-1 \right) \times \left( \sqrt[3]{{x}^{2}}+\sqrt[3]{x}+1 \right) \right)}^{2} }\right)$

To raise a product to a power, raise each factor to that power

$\lim_{x \rightarrow 1} \left(\frac{ {\left( \sqrt[3]{x}-1 \right)}^{2} }{ {\left( \sqrt[3]{x}-1 \right)}^{2} \times {\left( \sqrt[3]{{x}^{2}}+\sqrt[3]{x}+1 \right)}^{2} }\right)$

Cancel out the common factor ${\left( \sqrt[3]{x}-1 \right)}^{2}$

$\lim_{x \rightarrow 1} \left(\frac{ 1 }{ {\left( \sqrt[3]{{x}^{2}}+\sqrt[3]{x}+1 \right)}^{2} }\right)$

Evaluate the limit

$\frac{ 1 }{ {\left( \sqrt[3]{{1}^{2}}+\sqrt[3]{1}+1 \right)}^{2} }$

Simplify the expression

$\begin{align*}&\frac{ 1 }{ 9 } \\&\begin{array} { l }0.\overset{ \cdot }{ 1 } ,& {3}^{-2}\end{array}\end{align*}$