Question

$\underset{x\to \infty }{lim}\frac{{(x+1)}^{10}+{(x+2)}^{10}+....{(x+100)}^{10}}{x^{10}+{10}^{10}}$

• A. $0$
• B. $1$
• C. $100$
• D. $20$

Answer 1

Answer: (C)

$100$

$\underset{x\to \infty }{lim}\frac{(x^{10}{+}^{10}{C}_1{x}^9+....+1)+\left(x^{10}{+}^{10}{C}_1{x}^9\right(2)+....+1)+....\left(x^{10}{+}^{10}{C}_1{x}^9\right(100)+....+1)}{x^{10}+{10}^{10}}$

Taking $x^{10}$ common in both numerator and denominator

$=\underset{x\to \infty }{lim}{x}^{10}\frac{x^{10}(1{+}^{10}{C}_1\frac{1}{x}+...\frac{1}{x^{10}})+(1+\frac{{}^{10}{C}_1\left(2\right)}{x}+...+\frac{1}{x^{10}})+....(1+\frac{{}^{10}{C}_1\left(100\right)}{x}+...+\frac{1}{x^{10}})}{x^{10}(1+\frac{{10}^{10}}{x^{10}})}$

as $x\to \infty \frac{1}{x}\to 0$

$\frac{1}{x^n}\to 0$

$=\frac{(1+1+.....100times)}{1+0}=100$