o

\(lim_{x\rightarrow1}\frac{x^3+2x-3}{x^2-x}\)   

\(=lim_{x\rightarrow1}\frac{\left(x-1\right)\left(x^2+x+3\right)}{x\left(x-1\right)}\)   

\(=lim_{x\rightarrow1}\frac{x^2+x+3}{x}\)   

\(=\frac{1^2+1+3}{1}\)   

\(=5\)   

\(lim_{x\rightarrow1}\frac{\sqrt{2x+2}-\sqrt{3x+1}}{x-1}\)   

\(=lim_{x\rightarrow1}\frac{\left(2x+2\right)-\left(3x+1\right)}{\left(x-1\right)\left(\sqrt{2x+2}+\sqrt{3x+1}\right)}\)   

\(=lim_{x\rightarrow1}\frac{2x+2-3x-1}{\left(x-1\right)\left(\sqrt{2x+2}+\sqrt{3x+1}\right)}\)   

\(=lim_{x\rightarrow1}\frac{-x+1}{\left(x-1\right)\left(\sqrt{2x+2}+\sqrt{3x+1}\right)}\)   

\(=lim_{x\rightarrow1}\frac{-1\left(x-1\right)}{\left(x-1\right)\left(\sqrt{2x+2}+\sqrt{3x+1}\right)}\)   

\(=lim_{x\rightarrow1}\frac{-1}{\left(\sqrt{2x+2}+\sqrt{3x+1}\right)}\)   

\(=\frac{-1}{\sqrt{2\cdot1+2}+\sqrt{3\cdot1+1}}\)   

\(=\frac{-1}{2+2}=\frac{-1}{4}\)

https://drive.google.com/file/d/14Q-YI3szy-rePnIHWGD35RKCWiCXCT6k/view?usp=sharing

https://drive.google.com/file/d/1425SNt8hu4qt2y1kIcnhIvcxPfODsY1T/view?usp=sharing

\(\lim\limits_{x\rightarrow-2}=\dfrac{x-1+\sqrt{2x^2+1}}{4-x^2}\)

\(=\lim\limits_{x\rightarrow-2}=\dfrac{\left[\left(x-1\right)+\sqrt{2x^2+1}\right]\left[\left(x-1\right)-\sqrt{2x^2+1}\right]}{\left(4-x^2\right)\left[\left(x-1\right)-\sqrt{2x^2+1}\right]}\)

\(=\lim\limits_{x\rightarrow-2}\dfrac{\left(x-1\right)^2-\left(2x^2+1\right)}{\left(4-x^2\right)\left[\left(x-1\right)-\sqrt{2x^2+1}\right]}\)

\(=\lim\limits_{x\rightarrow-2}\dfrac{x^2-2x+1-2x^2-1}{\left(4-x^2\right)\left[\left(x-1\right)-\sqrt{2x^2+1}\right]}\)

\(=\lim\limits_{x\rightarrow-2}\dfrac{-x^2-2x}{\left(4-x^2\right)\left[\left(x-1\right)-\sqrt{2x^2+1}\right]}\)

\(=\lim\limits_{x\rightarrow-2}=-\dfrac{x}{\left(2-x\right)\left(x-1-\sqrt{2x^2+1}\right)}\)

\(=-\dfrac{1}{12}\)

\(\overset{lim}{x\rightarrow-2}\dfrac{x-1+\sqrt{2x^2+1}}{4-x^2}\) = \(\overset{lim}{x\rightarrow-2}\dfrac{x^2-2x+1-2x^2-1}{\left(4-x^2\right)\left(x-1-\sqrt{2x^2+1}\right)}\)

                                           = \(\overset{lim}{x\rightarrow-2}\dfrac{-x^2-2x}{\left(4-x^2\right)\left(x-1-\sqrt{2x^2+1}\right)}\)

                                          = \(\overset{lim}{x\rightarrow-2}\dfrac{-x\left(x+2\right)}{-\left(x-2\right)\left(x+2\right)\left(x-1-\sqrt{2x^2+1}\right)}\)

                                          = \(\overset{lim}{x\rightarrow-2}\dfrac{x}{\left(x-2\right)\left(x-1-\sqrt{2x^2+1}\right)}\)

                                          = \(\dfrac{-2}{\left(-2-2\right)\left[-2-1-\sqrt{2.\left(-2\right)^2+1}\right]}=-\dfrac{1}{12}\)

2

a: \(\lim\limits_{x\rightarrow-1^+}x+1=0\)

=>\(\lim\limits_{x\rightarrow-1^+}\dfrac{1}{x+1}=+\infty\)

b: \(\lim\limits_{x\rightarrow-\infty}1-x^2=\lim\limits_{x\rightarrow-\infty}\left[x^2\left(\dfrac{1}{x^2}-1\right)\right]\)

\(=-\infty\)

c: \(\lim\limits_{x\rightarrow3^-}\dfrac{x}{3-x}=\lim\limits_{x\rightarrow3^-}=\dfrac{-x}{x-3}\)

\(\lim\limits_{x\rightarrow3^-}x-3=0\)

\(\lim\limits_{x\rightarrow3^-}-x=3>0\)

=>\(\lim\limits_{x\rightarrow3^-}\dfrac{x}{3-x}=+\infty\)

a: \(=\lim\limits_{x\rightarrow+\infty}\dfrac{4+\dfrac{3}{x}}{2}=\dfrac{4}{2}=2\)

b: \(=\lim\limits_{x\rightarrow-\infty}\dfrac{\dfrac{2}{x}}{3+\dfrac{1}{x}}=0\)

c: \(=\lim\limits_{x\rightarrow+\infty}\dfrac{\sqrt{1+\dfrac{1}{x^2}}}{1+\dfrac{1}{x}}=1\)

a)-0,5

b)a=-3;b=2

a) lim⁡�→12�+3+�−5�−�2=lim⁡�→1(2�+3+(�−5))(2�+3−(�−5))(�−�2)(2�+3−(�−5))x→1lim​x−x22x+3​+x−5​=x→1lim​(x−x2)(2x+3​−(x−5))(2x+3​+(x−5))(2x+3​−(x−5))​

=lim⁡�→1−�2+14�−13−�(�−1)(2�+3−(�−5))=lim⁡�→1−(�−1)(�−13)−�(�−1)(2�+3−(�−5))=x→1lim​−x(x−1)(2x+3​−(x−5))−x2+14x−13​=x→1lim​−x(x−1)(2x+3​−(x−5))−(x−1)(x−13)​

=lim⁡�→1−(�−13)−�(2�+3−(�−5))=−32=x→1lim​−x(2x+3

a)-3/2.       b)a=-1, b=0

a) \(\mathop {\lim }\limits_{x \to 2} \left( {{x^2} - 4x + 3} \right) = \mathop {\lim }\limits_{x \to 2} {x^2} - \mathop {\lim }\limits_{x \to 2} \left( {4x} \right) + 3 = {2^2} - 4.2 + 3 =  - 1\)

b) \(\mathop {\lim }\limits_{x \to 3} \frac{{{x^2} - 5x + 6}}{{x - 3}} = \mathop {\lim }\limits_{x \to 3} \frac{{\left( {x - 3} \right)\left( {x - 2} \right)}}{{x - 3}} = \mathop {\lim }\limits_{x \to 3} \left( {x - 2} \right) = \mathop {\lim }\limits_{x \to 3} x - 2 = 3 - 2 = 1\)

c) \(\mathop {\lim }\limits_{x \to 1} \frac{{\sqrt x  - 1}}{{x - 1}} = \mathop {\lim }\limits_{x \to 1} \frac{{\sqrt x  - 1}}{{\left( {\sqrt x  - 1} \right)\left( {\sqrt x  + 1} \right)}} = \mathop {\lim }\limits_{x \to 1} \frac{1}{{\sqrt x  + 1}} = \frac{1}{{\sqrt 1  + 1}} = \frac{1}{2}\)

a) \(\mathop {\lim }\limits_{x \to  - \infty } \frac{{6x + 8}}{{5x - 2}} = \mathop {\lim }\limits_{x \to  - \infty } \frac{{x\left( {6 + \frac{8}{x}} \right)}}{{x\left( {5 - \frac{2}{x}} \right)}} = \frac{6}{5}\)

b) \(\mathop {\lim }\limits_{x \to  + \infty } \frac{{6x + 8}}{{5x - 2}} = \mathop {\lim }\limits_{x \to  + \infty } \frac{{x\left( {6 + \frac{8}{x}} \right)}}{{x\left( {5 - \frac{2}{x}} \right)}} = \mathop {\lim }\limits_{x \to  + \infty } \frac{{6 + \frac{8}{x}}}{{5 - \frac{2}{x}}} = \frac{6}{5}\).

c) \(\mathop {\lim }\limits_{x \to  - \infty } \frac{{\sqrt {9{x^2} - x + 1} }}{{3x - 2}} = \mathop {\lim }\limits_{x \to  - \infty } \frac{{ - x\sqrt {9 - \frac{1}{x} + \frac{1}{{{x^2}}}} }}{{x\left( {3 - \frac{2}{x}} \right)}} =  - \frac{3}{3} =  - 1\).

d) \(\mathop {\lim }\limits_{x \to  + \infty } \frac{{\sqrt {9{x^2} - x + 1} }}{{3x - 2}} = \mathop {\lim }\limits_{x \to  - \infty } \frac{{x\sqrt {9 - \frac{1}{x} + \frac{1}{{{x^2}}}} }}{{x\left( {3 - \frac{2}{x}} \right)}} = \frac{3}{3} = 1\).

e) \(\mathop {\lim }\limits_{x \to  - {2^ - }} \frac{{3{x^2} + 4}}{{2x + 4}} =  - \infty \)

Do \(\mathop {\lim }\limits_{x \to  - {2^ - }} \left( {3{x^2} + 1} \right) = 3.{\left( { - 2} \right)^2} + 1 = 13 > 0\) và \(\mathop {\lim }\limits_{x \to  - {2^ - }} \frac{1}{{2x + 4}} =  - \infty \)

g) \(\mathop {\lim }\limits_{x \to  - {2^ + }} \frac{{3{x^2} + 4}}{{2x + 4}} =  + \infty \).

Do \(\mathop {\lim }\limits_{x \to  - {2^ + }} \left( {3{x^2} + 1} \right) = 3.{\left( { - 2} \right)^2} + 1 = 13 > 0\) và \(\mathop {\lim }\limits_{x \to  - {2^ + }} \frac{1}{{2x + 4}} =  + \infty \)