Tag Archives: Limit

$f(x)=x^6+2x^4+x^3+2x+3 $
$\mathop {\lim }\limits_{x \to 1} \frac{{{x^n}f(1) – f(x)}}{{x – 1}} = 44$
$n=?$

June 21, 2022 Engineer Manish Verma Leave a comment

Let $f(x)=x^6+2x^4+x^3+2x+3,x \in R $. Then the natural number n for which $\mathop {\lim }\limits_{x \to 1} \frac{{{x^n}f(1) – f(x)}}{{x – 1}} = 44$ is _ _ _ _ .

Solution

Since the limit has $\left[ {\frac{0}{0}} \right]$ form, L.H. Rule is applicable.

Thus, $\mathop {\lim }\limits_{x \to 1} n{x^{n – 1}}f(1) – f'(x) = 44$

$\therefore nf(1) – f'(1) = 44$

$\therefore n.9 – ({6.1^5} + {8.1^3} + {3.1^2} + 2.1) = 44$

$ \Rightarrow 9n – 19 = 44$

$\Rightarrow n=7$

Mathematics

Let $\alpha (a)$ and $\beta (a)$ be the roots ………

December 5, 2014 Engineer Manish Verma Leave a comment

Let $\alpha (a)$ and $\beta (a)$ be the roots of the equation

$\left( {\sqrt[3]{{1 + a}} - 1} \right){x^2} + \left( {\sqrt {1 + a} - 1} \right)x + \left( {\sqrt[6]{{1 + a}} - 1} \right) = 0$ where a>-1.

Then ${\lim _{a \to 0 + }}\alpha (a)$ and ${\lim _{a \to 0 + }}\beta (a)$ are

(A) $- \frac{5}{2}$ and 1 (B) $- \frac{1}{2}$ and -1 (C) $- \frac{7}{2}$ and 2 (D) $- \frac{9}{2}$ and 3

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[JEE 2012]

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Tag Archives: Limit

$\mathop {\lim }\limits_{x \to \infty} \frac {x^3}{\sqrt {x^2-a^2}}-\frac {x^3}{\sqrt {x^2+a^2}}=?$

$f(x)=x^6+2x^4+x^3+2x+3 $
$\mathop {\lim }\limits_{x \to 1} \frac{{{x^n}f(1) – f(x)}}{{x – 1}} = 44$
$n=?$

Let \(\alpha (a)\) and \(\beta (a)\) be the roots ………

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