Question

lim x-7 [x³-343] /√x-√7



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Answer 1

Given limit,

$\displaystyle\longrightarrow\lim_{x\to 7}\left(\dfrac{x^3-343}{\sqrt x-\sqrt7}\right)=\lim_{x\to 7}\left(\dfrac{x^3-7^3}{x^{\frac{1}{2}}-7^{\frac{1}{2}}}\right)$

We have,

$\displaystyle\longrightarrow \lim_{x\to a}\left(\dfrac{x^n-a^n}{x-a}\right)=na^{n-1}$

Using this formula we can have the following result.

$\displaystyle\longrightarrow \lim_{x\to a}\left(\dfrac{x^m-a^m}{x-a}\right)\div\lim_{x\to a}\left(\dfrac{x^n-a^n}{x-a}\right)=\dfrac{ma^{m-1}}{na^{n-1}}$

$\displaystyle\longrightarrow \lim_{x\to a}\left(\dfrac{x^m-a^m}{x-a}\div\dfrac{x^n-a^n}{x-a}\right)=\dfrac{m}{n}\cdot a^{(m-1)-(n-1)}$

$\displaystyle\longrightarrow \lim_{x\to a}\left(\dfrac{x^m-a^m}{x-a}\times\dfrac{x-a}{x^n-a^n}\right)=\dfrac{m}{n}\cdot a^{m-1-n+1}$

$\displaystyle\longrightarrow \lim_{x\to a}\left(\dfrac{x^m-a^m}{x^n-a^n}\right)=\dfrac{m}{n}\cdot a^{m-n}$

Taking $a=7,\ m=3$ and $n=\dfrac{1}{2},$

$\displaystyle\longrightarrow \lim_{x\to 7}\left(\dfrac{x^3-7^3}{x^{\frac{1}{2}}-7^{\frac{1}{2}}}\right)=\dfrac{3}{\left(\dfrac{1}{2}\right)}\cdot 7^{\,3-\frac{1}{2}}$

$\displaystyle\longrightarrow\underline{\underline{\lim_{x\to 7}\left(\dfrac{x^3-343}{\sqrt x-\sqrt7}\right)=6\times7^{\frac{5}{2}}}}$