# Calculate lim_(x ->x_o) (x^2- x_o x)/(x^2-x_o^2) for every value of x_o in RR ?

Feb 18, 2018

lim_(x→x_0) (x^2-x_0x)/(x^2-x_0^2) = 1/2

#### Explanation:

lim_(x→x_0) (x^2-x_0x)/(x^2-x_0^2) = lim_(x→x_0) (x(x-x_0))/((x-x_0)(x+x_0))=lim_(x→x_0) x/(x+x_0)

We can now let $x = {x}_{0}$ to get the limit as

${x}_{0} / \left({x}_{0} + {x}_{0}\right) = {x}_{0} / \left(2 {x}_{0}\right) = \frac{1}{2}$

Feb 18, 2018

${\lim}_{x \to {x}_{0}} \frac{{x}^{2} - {x}_{0} x}{{x}^{2} - {x}_{0}^{2}} = \left\{\begin{matrix}\frac{1}{2} \text{ for " x_0 !=0 \\ 1 " for } {x}_{0} = 0\end{matrix}\right.$

#### Explanation:

Simplify the function:

$\frac{{x}^{2} - {x}_{0} x}{{x}^{2} - {x}_{0}^{2}} = \frac{x \left(x - {x}_{0}\right)}{\left(x + {x}_{0}\right) \left(x - {x}_{0}\right)} = \frac{x}{x + {x}_{0}}$

so for ${x}_{0} \ne 0$:

${\lim}_{x \to {x}_{0}} \frac{{x}^{2} - {x}_{0} x}{{x}^{2} - {x}_{0}^{2}} = {\lim}_{x \to {x}_{0}} \frac{x}{x + {x}_{0}} = {x}_{0} / \left({x}_{0} + {x}_{0}\right) = \frac{1}{2}$

while for ${x}_{0} = 0$:

${\lim}_{x \to {x}_{0}} \frac{{x}^{2} - {x}_{0} x}{{x}^{2} - {x}_{0}^{2}} = {\lim}_{x \to 0} {x}^{2} / {x}^{2} = 1$