Ответ:

[tex] \frac{lim}{x - > 1}( \frac{ {4x}^{2} + 3x - 1}{ {x}^{2} + 2x + 1 } )[/tex]

[tex] \frac{ \frac{lim}{x - > 1}( {4x}^{2} + 3x - 1)}{ \frac{lim}{x - > 1} ( {x}^{2} + 2x + 1)} [/tex]

[tex] \frac{ \frac{lim}{x - > 1}( {4x}^{2} + 3x) - \frac{lim}{x - > 1} (1) }{ \frac{lim}{x - > 1} ( {x}^{2} + 2x) + \frac{lim}{x - > 1}(1) } [/tex]

[tex] \frac{ \frac{lim}{x - > 1}( {4x}^{2}) + \frac{lim}{x - > 1}(3x) - 1 }{ \frac{lim}{x - > 1} ( {x}^{2} ) + \frac{lim}{x - > 1}(2x) + 1 } [/tex]

[tex] \frac{4 \times \frac{lim}{x - > 1} ( {x}^{2}) + 3 \times \frac{lim}{x - > 1} (x) - 1 }{ {( \frac{lim}{x - > 1} (x))}^{2} + 2 \times \frac{lim}{x - > 1} (x) + 1 } [/tex]

[tex] \frac{4 \times {( \frac{lim}{x - > 1} (x))}^{2} + 3 \times 1 - 1 }{ {1}^{2} + 2 \times 1 + 1} [/tex]

[tex] \frac{4 \times {1}^{2} + 3 \times 1 - 1 }{ {1}^{2} + 2 \times 1 + 1} [/tex]

[tex] \frac{4 \times 1 + 3 - 1}{1 + 2 + 1} [/tex]

[tex] \frac{4 + 3 - 1}{1 + 2 + 1} [/tex]

[tex] \frac{6}{4} [/tex]

сокращаем на общий делитель 2:

[tex] \frac{3}{2} [/tex]