mathop {lim }limits_{x to infty } frac{{{{(x | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) We are given the limit: $\mathop {\lim }\limits_{x 	o \infty } \frac{{{{(x + 1)}^{10}} + {{(x + 2)}^{10}} + \dots + {{(x + 100)}^{10}}}}{{{x^{10}

Vedclass · Accepted Answer

$100$

Vedclass · Answer

(D) We are given the limit: $\mathop {\lim }\limits_{x 	o \infty } \frac{{{{(x + 1)}^{10}} + {{(x + 2)}^{10}} + \dots + {{(x + 100)}^{10}}}}{{{x^{10}} + {{10}^{10}}}}$Divide the numerator and denominator by $x^{10}$:$= \mathop {\lim }\limits_{x 	o \infty } \frac{{{{\left( {1 + \frac{1}{x}} ight)}^{10}} + {{\left( {1 + \frac{2}{x}} ight)}^{10}} + \dots + {{\left( {1 + \frac{{100}}{x}} ight)}^{10}}}}{{1 + \frac{{{{10}^{10}}}}{{{x^{10}}}}}}$As $x 	o \infty$,the terms $\frac{k}{x} 	o 0$ for any constant $k$,and $\frac{10^{10}}{x^{10}} 	o 0$.Therefore,the expression becomes:$= \frac{{{{(1 + 0)}^{10}} + {{(1 + 0)}^{10}} + \dots + {{(1 + 0)}^{10}}}}{{1 + 0}}$Since there are $100$ terms in the numerator,each equal to $1^{10} = 1$:$= \frac{1 + 1 + \dots + 1 	ext{ (100 times)}}{1} = 100$.

$\mathop {\lim }\limits_{x \to \infty } \frac{{{{(x + 1)}^{10}} + {{(x + 2)}^{10}} + \dots + {{(x + 100)}^{10}}}}{{{x^{10}} + {{10}^{10}}}}$ is equal to

Similar Questions

$\lim _{x \rightarrow \infty}\left(\frac{x+5}{x+2}\right)^{x+3}$ equals

$\lim _{x \rightarrow 0}\left(\frac{1+\tan x}{1+\sin x}\right)^{\operatorname{cosec} x}=$

$\lim _{n \rightarrow \infty}\left(1+\frac{1+\frac{1}{2}+\ldots+\frac{1}{n}}{n^{2}}\right)^{n} = \dots$

$\mathop {\lim }\limits_{n \to \infty } {\left( {\frac{n}{{n + y}}} \right)^n}$ equals

If $\lim _{x \rightarrow 0^{+}} x^2\left(\frac{e^{1 / x}-e^{-1 / x}}{e^{1 / x}+e^{-1 / x}}\right)=k$ and $\lim _{x \rightarrow 0^{-}} x^2\left(\frac{e^{1 / x}-e^{-1 / x}}{e^{1 / x}+e^{-1 / x}}\right)=l$,then which of the following is true?

Vedclass Test Series

Exam Paper Generator

Online Exam Module