Solve for: \lim_{x arrow 3} ((x^2-6x+9)/(x^2-9))

Expression: $\lim_{x \rightarrow 3} \left(\frac{ {x}^{2}-6x+9 }{ {x}^{2}-9 }\right)$

Since evaluating limits of the numerator and denominator would result in an indeterminate form, use the L'Hopital's rule

$\lim_{x \rightarrow 3} \left(\frac{ \frac{ \mathrm{d} }{ \mathrm{d}x} \left( {x}^{2}-6x+9 \right) }{ \frac{ \mathrm{d} }{ \mathrm{d}x} \left( {x}^{2}-9 \right) }\right)$

Find the derivative

$\lim_{x \rightarrow 3} \left(\frac{ 2x-6 }{ \frac{ \mathrm{d} }{ \mathrm{d}x} \left( {x}^{2}-9 \right) }\right)$

Find the derivative

$\lim_{x \rightarrow 3} \left(\frac{ 2x-6 }{ 2x }\right)$

Factor out $2$ from the expression

$\lim_{x \rightarrow 3} \left(\frac{ 2\left( x-3 \right) }{ 2x }\right)$

Cancel out the common factor $2$

$\lim_{x \rightarrow 3} \left(\frac{ x-3 }{ x }\right)$

Evaluate the limit

$\frac{ 3-3 }{ 3 }$

Simplify the expression

$0$