Đáp án:

\(\begin{array}{l}
a)\quad \lim\limits_{x\to 2}\dfrac{3x^2 + 2x -5}{x+2}=\dfrac{11}{4}\\
b)\quad \lim\limits_{x\to -1}\dfrac{x^3 - x^2 + 2x +4}{-x^2 + 4x+5}=\dfrac76\\
c)\quad \lim\limits_{x\to - \infty}\left(\sqrt{3x^2 + 5x - 4} + \sqrt3x\right)=-\dfrac{5\sqrt3}{6}\\
d)\quad \lim\limits_{x\to 7}\dfrac{2 - \sqrt{x-3}}{x^2- 49}=-\dfrac{1}{56}
\end{array}\)

Giải thích các bước giải:

\(\begin{array}{l}
a)\quad \lim\limits_{x\to 2}\dfrac{3x^2 + 2x -5}{x+2}\\
= \dfrac{3.2^2 + 2.2 - 5}{2+2}\\
= \dfrac{11}{4}\\
b)\quad \lim\limits_{x\to -1}\dfrac{x^3 - x^2 + 2x +4}{-x^2 + 4x+5}\\
= \lim\limits_{x\to -1}\dfrac{(x+1)(x^2 - 2x + 4)}{(x+1)(5-x)}\\
=\lim\limits_{x\to -1}\dfrac{x^2 - 2x + 4}{5-x}\\
= \dfrac{1 + 2 + 4}{5 + 1}\\
= \dfrac76\\
c)\quad \lim\limits_{x\to - \infty}\left(\sqrt{3x^2 + 5x - 4} + \sqrt3x\right)\\
=\lim\limits_{x\to - \infty}\dfrac{\left(\sqrt{3x^2 + 5x - 4} + \sqrt3x\right)\left(\sqrt{3x^2 + 5x - 4} - \sqrt3x\right)}{\sqrt{3x^2 + 5x - 4} - \sqrt3x}\\
=\lim\limits_{x\to - \infty}\dfrac{5x - 4}{\sqrt{3x^2 + 5x - 4} - \sqrt3x}\\
=\lim\limits_{x\to - \infty}\dfrac{5 - \dfrac4x}{-\sqrt{3 + \dfrac5x - \dfrac{4}{x^2}} - \sqrt3}\\
= \dfrac{5-0}{-\sqrt{3+0-0} -\sqrt3}\\
= -\dfrac{5\sqrt3}{6}\\
d)\quad \lim\limits_{x\to 7}\dfrac{2 - \sqrt{x-3}}{x^2- 49}\\
=\lim\limits_{x\to 7}\dfrac{\left(2 - \sqrt{x-3}\right)\left(2+ \sqrt{x-3}\right)}{(x-7)(x+7)\left(2 + \sqrt{x-3}\right)}\\
=\lim\limits_{x\to 7}\dfrac{7-x}{(x-7)(x+7)\left(2 + \sqrt{x-3}\right)}\\
= \lim\limits_{x\to 7}\dfrac{-1}{(x+7)\left(2 + \sqrt{x-3}\right)}\\
= \dfrac{-1}{(7+7)\left(2+\sqrt{7-3}\right)}\\
=-\dfrac{1}{56}
\end{array}\) 

Đáp án:

Giải thích các bước giải:

 `a, lim_{x->2}\frac{3x²+2x-5}{x+2}`

`= \frac{3.2²+2.2-5}{2+2}`

`= \frac{11}{4}`

`b, lim_{x->-1}\frac{x³-x²+2x+4}{-x²+4x+5}`

`= lim_{x->-1} \frac{(x+1)(x²-2x+4)}{-(x+1)(x-5)}`

`= lim_{x->-1}-\frac{x²-2x+4}{x-5}`

`= -\frac{(-1)²-2.(-1)+4}{-1-5}`

`=7/6`

`c, lim_{x->-\infty}\sqrt{3x²+5x-4}+\sqrt{3}x`

` =lim_{x->-\infty}\frac{3x²+5x-4-3x²}{\sqrt{3x²+5x-4}+\sqrt{3}x`

`=lim_{x->-\infty} \frac{5x-4}{\sqrt{3x²+5x-4}+\sqrt{3}x`

`=lim_{x->-\infty} \frac{x(5-4/x)}{x(-\sqrt{3+5/x-4/{x²}}+\sqrt{3})`

`=lim_{x->-\infty} \frac{5-4/x}{-\sqrt{3+5/x-4/{x²}}+\sqrt{3}`

`= -\frac{5\sqrt{3}}{6}`  

`d, lim_{x->7}\frac{2-\sqrt{x-3}}{x²-49}`

`= lim_{x->7}\frac{4-x+3}{(x-7)(x+7)(2+\sqrt{x+3})`

`= lim_{x->7}\frac{7-x}{(x-7)(x+7)(2+\sqrt{x+3})`

`= lim_{x->7}\frac{-1}{(x+7)(2+\sqrt{x+3})`

`= \frac{-1}{(7+7)(2+\sqrt{7+3})`

`= -1/{56}`