Câu 2c:
\[\begin{align} & \underset{x\to 1}{\mathop{\lim }}\,\frac{\sqrt{x+3}+{{x}^{2}}+x-4}{x-1} \\ & =\underset{x\to 1}{\mathop{\lim }}\,\frac{\sqrt{x+3}-2+{{x}^{2}}+x-2}{x-1} \\ & =\underset{x\to 1}{\mathop{\lim }}\,\frac{\sqrt{x+3}-2}{x-1}+\frac{{{x}^{2}}+x-2}{x-1} \\ & =\underset{x\to 1}{\mathop{\lim }}\,\frac{x-1}{(x-1)(\sqrt{x+3}+2)}+\frac{(x-1)(x+2)}{x-1} \\ & =\underset{x\to 1}{\mathop{\lim }}\,\frac{1}{\sqrt{x+3}+2}+\frac{x+2}{1} \\ & =\frac{1}{\sqrt{1+3}+2}+\frac{1+2}{1}=\frac{13}{4} \\ \end{align}\]
câu 2d:
\[\begin{align} & \underset{x\to 1}{\mathop{\lim }}\,\frac{\sqrt{2x-1}-\sqrt[3]{3{{x}^{2}}-3x+1}}{{{(x-1)}^{2}}} \\ & =\underset{x\to 1}{\mathop{\lim }}\,\frac{(\sqrt{2x-1}-x)+(x-\sqrt[3]{3{{x}^{2}}-3x+1})}{{{(x-1)}^{2}}} \\ & =\underset{x\to 1}{\mathop{\lim }}\,\frac{\sqrt{2x-1}-x}{{{(x-1)}^{2}}}+\frac{x-\sqrt[3]{3{{x}^{2}}-3x+1}}{{{(x-1)}^{2}}} \\ & =\underset{x\to 1}{\mathop{\lim }}\,\frac{-{{x}^{2}}+2x-1}{{{(x-1)}^{2}}(\sqrt{2x-1}+x)}+\frac{{{x}^{3}}-3{{x}^{2}}+3x-1}{{{(x-1)}^{2}}({{x}^{2}}+x\sqrt[3]{3{{x}^{2}}-3x+1}+\sqrt[3]{{{(3{{x}^{2}}-3x+1)}^{2}}})} \\ & =\underset{x\to 1}{\mathop{\lim }}\,\frac{{{-(x-1)}^{2}}}{{{(x-1)}^{2}}(\sqrt{2x-1}+x)}+\frac{{{(x-1)}^{3}}}{{{(x-1)}^{2}}({{x}^{2}}+x\sqrt[3]{3{{x}^{2}}-3x+1}+\sqrt[3]{{{(3{{x}^{2}}-3x+1)}^{2}}})} \\ & =\underset{x\to 1}{\mathop{\lim }}\,\frac{-1}{\sqrt{2x-1}+x}+\frac{x-1}{{{x}^{2}}+x\sqrt[3]{3{{x}^{2}}-3x+1}+\sqrt[3]{{{(3{{x}^{2}}-3x+1)}^{2}}}} \\ & =\frac{-1}{\sqrt{2-1}+1}+\frac{1-1}{{{1}^{2}}+\sqrt[3]{3-3+1}+\sqrt[3]{{{(3-3+1)}^{2}}}}=-\frac{1}{2} \\ \end{align}\]
Câu 12:
\[\begin{align} & \underset{x\to \frac{\pi }{4}}{\mathop{\lim }}\,\frac{\sin x-\cos x}{\tan \left( \frac{\pi }{4}-x \right)}=\underset{x\to \frac{\pi }{4}}{\mathop{\lim }}\,(\sin x-\cos x).\left[ \frac{c\text{os}\left( \frac{\pi }{4}-x \right)}{\sin \left( \frac{\pi }{4}-x \right)} \right] \\ & =\underset{x\to \frac{\pi }{4}}{\mathop{\lim }}\,(\sin x-\cos x).\left( \frac{c\text{os}\frac{\pi }{4}.\cos x+\sin \frac{\pi }{4}.\sin x}{\sin \frac{\pi }{4}.\cos x-c\text{os}\frac{\pi }{4}.\sin x} \right) \\ & =\underset{x\to \frac{\pi }{4}}{\mathop{\lim }}\,(\sin x-\cos x).\left( \frac{\frac{\sqrt{2}}{2}\cos x+\frac{\sqrt{2}}{2}.\sin x}{\frac{\sqrt{2}}{2}\cos x-\frac{\sqrt{2}}{2}\sin x} \right) \\ & =\underset{x\to \frac{\pi }{4}}{\mathop{\lim }}\,(\sin x-\cos x).\left( \frac{\frac{\sqrt{2}}{2}(\cos x+\sin x)}{\frac{\sqrt{2}}{2}(\cos x-\sin x)} \right) \\ & =\underset{x\to \frac{\pi }{4}}{\mathop{\lim }}\,\frac{(\sin x-\cos x).(\sin x+\cos x)}{-(\sin x-\cos x)} \\ & =\underset{x\to \frac{\pi }{4}}{\mathop{\lim }}\,-(\sin x+\cos x) \\ & =-\left( \sin \frac{\pi }{4}+\cos \frac{\pi }{4} \right)=-\left( \frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2} \right)=-\sqrt{2} \\ \end{align}\]
