# Evaluate: \lim_{u arrow 1} ((u^2-ux+2u-2x)/(u^2-u-6))

## Expression: $\lim_{u \rightarrow 1} \left(\frac{ {u}^{2}-ux+2u-2x }{ {u}^{2}-u-6 }\right)$

Use $\lim_{x \rightarrow c} \left(\frac{ f\left( x \right) }{ g\left( x \right) }\right)=\frac{ \lim_{x \rightarrow c} \left(f\left( x \right)\right) }{ \lim_{x \rightarrow c} \left(g\left( x \right)\right) }$ to transform the expression

$\frac{ \lim_{u \rightarrow 1} \left({u}^{2}-ux+2u-2x\right) }{ \lim_{u \rightarrow 1} \left({u}^{2}-u-6\right) }$

Use $\lim_{x \rightarrow c} \left(\left( f\left( x \right)\pmg\left( x \right) \right)\right)=\lim_{x \rightarrow c} \left(f\left( x \right)\right)\pm\lim_{x \rightarrow c} \left(g\left( x \right)\right)$ to transform the expression

$\frac{ \lim_{u \rightarrow 1} \left({u}^{2}-ux+2u\right)-\lim_{u \rightarrow 1} \left(2x\right) }{ \lim_{u \rightarrow 1} \left({u}^{2}-u-6\right) }$

Use $\lim_{x \rightarrow c} \left(\left( f\left( x \right)\pmg\left( x \right) \right)\right)=\lim_{x \rightarrow c} \left(f\left( x \right)\right)\pm\lim_{x \rightarrow c} \left(g\left( x \right)\right)$ to transform the expression

$\frac{ \lim_{u \rightarrow 1} \left({u}^{2}-ux+2u\right)-\lim_{u \rightarrow 1} \left(2x\right) }{ \lim_{u \rightarrow 1} \left({u}^{2}-u\right)-\lim_{u \rightarrow 1} \left(6\right) }$

Use $\lim_{x \rightarrow c} \left(\left( f\left( x \right)\pmg\left( x \right) \right)\right)=\lim_{x \rightarrow c} \left(f\left( x \right)\right)\pm\lim_{x \rightarrow c} \left(g\left( x \right)\right)$ to transform the expression

$\frac{ \lim_{u \rightarrow 1} \left({u}^{2}-ux\right)+\lim_{u \rightarrow 1} \left(2u\right)-\lim_{u \rightarrow 1} \left(2x\right) }{ \lim_{u \rightarrow 1} \left({u}^{2}-u\right)-\lim_{u \rightarrow 1} \left(6\right) }$

The limit of a constant is equal to the constant

$\frac{ \lim_{u \rightarrow 1} \left({u}^{2}-ux\right)+\lim_{u \rightarrow 1} \left(2u\right)-2x }{ \lim_{u \rightarrow 1} \left({u}^{2}-u\right)-\lim_{u \rightarrow 1} \left(6\right) }$

Use $\lim_{x \rightarrow c} \left(\left( f\left( x \right)\pmg\left( x \right) \right)\right)=\lim_{x \rightarrow c} \left(f\left( x \right)\right)\pm\lim_{x \rightarrow c} \left(g\left( x \right)\right)$ to transform the expression

$\frac{ \lim_{u \rightarrow 1} \left({u}^{2}-ux\right)+\lim_{u \rightarrow 1} \left(2u\right)-2x }{ \lim_{u \rightarrow 1} \left({u}^{2}\right)-\lim_{u \rightarrow 1} \left(u\right)-\lim_{u \rightarrow 1} \left(6\right) }$

The limit of a constant is equal to the constant

$\frac{ \lim_{u \rightarrow 1} \left({u}^{2}-ux\right)+\lim_{u \rightarrow 1} \left(2u\right)-2x }{ \lim_{u \rightarrow 1} \left({u}^{2}\right)-\lim_{u \rightarrow 1} \left(u\right)-6 }$

Use $\lim_{x \rightarrow c} \left(\left( f\left( x \right)\pmg\left( x \right) \right)\right)=\lim_{x \rightarrow c} \left(f\left( x \right)\right)\pm\lim_{x \rightarrow c} \left(g\left( x \right)\right)$ to transform the expression

$\frac{ \lim_{u \rightarrow 1} \left({u}^{2}\right)-\lim_{u \rightarrow 1} \left(ux\right)+\lim_{u \rightarrow 1} \left(2u\right)-2x }{ \lim_{u \rightarrow 1} \left({u}^{2}\right)-\lim_{u \rightarrow 1} \left(u\right)-6 }$

Use $\lim_{x \rightarrow c} \left(a \times f\left( x \right)\right)=a \times \lim_{x \rightarrow c} \left(f\left( x \right)\right)$ to transform the expression

$\frac{ \lim_{u \rightarrow 1} \left({u}^{2}\right)-\lim_{u \rightarrow 1} \left(ux\right)+2 \times \lim_{u \rightarrow 1} \left(u\right)-2x }{ \lim_{u \rightarrow 1} \left({u}^{2}\right)-\lim_{u \rightarrow 1} \left(u\right)-6 }$

Use $\lim_{x \rightarrow c} \left({f\left( x \right)}^{a}\right)={\left( \lim_{x \rightarrow c} \left(f\left( x \right)\right) \right)}^{a}$ to transform the expression

$\frac{ \lim_{u \rightarrow 1} \left({u}^{2}\right)-\lim_{u \rightarrow 1} \left(ux\right)+2 \times \lim_{u \rightarrow 1} \left(u\right)-2x }{ {\left( \lim_{u \rightarrow 1} \left(u\right) \right)}^{2}-\lim_{u \rightarrow 1} \left(u\right)-6 }$

Evaluate the limit by substituting the value $u=1$ into the expression

$\frac{ \lim_{u \rightarrow 1} \left({u}^{2}\right)-\lim_{u \rightarrow 1} \left(ux\right)+2 \times \lim_{u \rightarrow 1} \left(u\right)-2x }{ {\left( \lim_{u \rightarrow 1} \left(u\right) \right)}^{2}-1-6 }$

Use $\lim_{x \rightarrow c} \left({f\left( x \right)}^{a}\right)={\left( \lim_{x \rightarrow c} \left(f\left( x \right)\right) \right)}^{a}$ to transform the expression

$\frac{ {\left( \lim_{u \rightarrow 1} \left(u\right) \right)}^{2}-\lim_{u \rightarrow 1} \left(ux\right)+2 \times \lim_{u \rightarrow 1} \left(u\right)-2x }{ {\left( \lim_{u \rightarrow 1} \left(u\right) \right)}^{2}-1-6 }$

Use $\lim_{x \rightarrow c} \left(a \times f\left( x \right)\right)=a \times \lim_{x \rightarrow c} \left(f\left( x \right)\right)$ to transform the expression

$\frac{ {\left( \lim_{u \rightarrow 1} \left(u\right) \right)}^{2}-x \times \lim_{u \rightarrow 1} \left(u\right)+2 \times \lim_{u \rightarrow 1} \left(u\right)-2x }{ {\left( \lim_{u \rightarrow 1} \left(u\right) \right)}^{2}-1-6 }$

Evaluate the limit by substituting the value $u=1$ into the expression

$\frac{ {\left( \lim_{u \rightarrow 1} \left(u\right) \right)}^{2}-x \times \lim_{u \rightarrow 1} \left(u\right)+2 \times 1-2x }{ {\left( \lim_{u \rightarrow 1} \left(u\right) \right)}^{2}-1-6 }$

Evaluate the limit by substituting the value $u=1$ into the expression

$\frac{ {\left( \lim_{u \rightarrow 1} \left(u\right) \right)}^{2}-x \times \lim_{u \rightarrow 1} \left(u\right)+2 \times 1-2x }{ {1}^{2}-1-6 }$

Evaluate the limit by substituting the value $u=1$ into the expression

$\frac{ {1}^{2}-x \times \lim_{u \rightarrow 1} \left(u\right)+2 \times 1-2x }{ {1}^{2}-1-6 }$

Evaluate the limit by substituting the value $u=1$ into the expression

$\frac{ {1}^{2}-x \times 1+2 \times 1-2x }{ {1}^{2}-1-6 }$

Simplify the expression

$\frac{ x-1 }{ 2 }$

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