Question:

$\displaystyle\lim_{n\to\infty} \frac{\left(1^{2} +2^{2} + ....+n^{2}\right) \sqrt[n]{n}}{ \left(n+1\right)\left(n+10\right)\left(n+100\right)} = $

Updated On: May 11, 2024
  • $3$
  • $1/3$
  • $2/3$
  • $ \infty$
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The Correct Option is B

Solution and Explanation

$\displaystyle\lim_{n\to\infty} \frac{n\left(n+1\right)\left(2n+1\right)\sqrt[n]{n}}{6\left(n+1\right)\left(n+10\right)\left(n+100\right)}$
$ = \displaystyle\lim _{n\to \infty } \frac{\left(2+\frac{1}{n}\right) \displaystyle\lim _{n\to \infty } \sqrt[n]{n}}{6\left(1+\frac{10}{n}\right)\left(1+\frac{100}{n}\right)} = \frac{2}{6} = \frac{1}{3}$
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Concepts Used:

Limits And Derivatives

Mathematically, a limit is explained as a value that a function approaches as the input, and it produces some value. Limits are essential in calculus and mathematical analysis and are used to define derivatives, integrals, and continuity.

Limit of a Function

Limits Formula:

Limits Formula
 Derivatives of a Function:

derivative is referred to the instantaneous rate of change of a quantity with response to the other. It helps to look into the moment-by-moment nature of an amount. The derivative of a function is shown in the below-given formula.

 Derivatives of a Function

Properties of Derivatives:

Properties of Derivatives

Read More: Limits and Derivatives