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