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Câu hỏi: Cho $\underset{x\to 0}{\mathop{\lim }} \left( \dfrac{x}{\sqrt[7]{x+1}.\sqrt{x+4}-2} \right)=\dfrac{a}{b}$ ( $\dfrac{a}{b}$ là phân số tối giản). Tính tổng $L=a+b$.
A. $L=43$.
B. $L=23$.
C. $L=13$.
D. $L=53$.
A. $L=43$.
B. $L=23$.
C. $L=13$.
D. $L=53$.
Ta xét $T=\underset{x\to 0}{\mathop{\lim }} \left( \dfrac{\sqrt[7]{x+1}.\sqrt{x+4}-2}{x} \right)$
$T=\underset{x\to 0}{\mathop{\lim }} \left( \dfrac{\sqrt[7]{x+1}.\left( \sqrt{x+4}-2 \right)+2\left( \sqrt[7]{x+1}-1 \right)}{x} \right)$
Ta xét ${{T}_{1}}=\underset{x\to 0}{\mathop{\lim }} \dfrac{\sqrt[7]{x+1}\left( x+4-4 \right)}{x\left( \sqrt{x+4}+2 \right)}=\underset{x\to 0}{\mathop{\lim }} \dfrac{\sqrt[7]{x+1}}{\sqrt{x+4}+2}=\dfrac{1}{4}.$
Ta xét ${{T}_{2}}=\underset{x\to 0}{\mathop{\lim }} \dfrac{2\left( \sqrt[7]{x+1}-1 \right)}{x}.$
Đặt $t=\sqrt[7]{x+1}\Rightarrow x={{t}^{7}}-1.$
Ta có: $x\to 0\Rightarrow t\to 1.$ Khi đó, ${{T}_{2}}=\underset{t\to 1}{\mathop{\lim }} \dfrac{2\left( t-1 \right)}{{{t}^{7}}-1}=\underset{t\to 1}{\mathop{\lim }} \dfrac{2}{{{t}^{6}}+{{t}^{5}}+{{t}^{4}}+{{t}^{3}}+{{t}^{2}}+t+1}=\dfrac{2}{7}.$
$\Rightarrow T={{T}_{1}}+{{T}_{2}}=\dfrac{1}{4}+\dfrac{2}{7}=\dfrac{15}{28}.$
Vậy $a=28, b=15.$ Do đó, $L=a+b=28+15=43.$
$T=\underset{x\to 0}{\mathop{\lim }} \left( \dfrac{\sqrt[7]{x+1}.\left( \sqrt{x+4}-2 \right)+2\left( \sqrt[7]{x+1}-1 \right)}{x} \right)$
Ta xét ${{T}_{1}}=\underset{x\to 0}{\mathop{\lim }} \dfrac{\sqrt[7]{x+1}\left( x+4-4 \right)}{x\left( \sqrt{x+4}+2 \right)}=\underset{x\to 0}{\mathop{\lim }} \dfrac{\sqrt[7]{x+1}}{\sqrt{x+4}+2}=\dfrac{1}{4}.$
Ta xét ${{T}_{2}}=\underset{x\to 0}{\mathop{\lim }} \dfrac{2\left( \sqrt[7]{x+1}-1 \right)}{x}.$
Đặt $t=\sqrt[7]{x+1}\Rightarrow x={{t}^{7}}-1.$
Ta có: $x\to 0\Rightarrow t\to 1.$ Khi đó, ${{T}_{2}}=\underset{t\to 1}{\mathop{\lim }} \dfrac{2\left( t-1 \right)}{{{t}^{7}}-1}=\underset{t\to 1}{\mathop{\lim }} \dfrac{2}{{{t}^{6}}+{{t}^{5}}+{{t}^{4}}+{{t}^{3}}+{{t}^{2}}+t+1}=\dfrac{2}{7}.$
$\Rightarrow T={{T}_{1}}+{{T}_{2}}=\dfrac{1}{4}+\dfrac{2}{7}=\dfrac{15}{28}.$
Vậy $a=28, b=15.$ Do đó, $L=a+b=28+15=43.$
Đáp án A.