displaystyle lim x arrow ∞ ((x+1)10+(x+2)10+ ldots .+(x+100)10/x10+1010) is equal to

Question

displaystyle lim x arrow ∞ ((x+1)10+(x+2)10+ ldots .+(x+100)10/x10+1010) is equal to (A) 0 (B) 1 (C) 10 (D) 100

Tardigrade Admin · Accepted Answer

displaystyle lim x arrow ∞ ((x+1)10+(x+2)10+ ldots .+(x+100)10/x10+1010) = displaystyle lim x arrow ∞ (x10[(1+(1/x))10+(1+(2/x))10+ ldots+(1+(100/x))10]/x10[1+(1010)x10])=100.

Q. $x \to \infty lim \frac{( x + 1 ) ^{10} + ( x + 2 ) ^{10} + \dots . + ( x + 100 ) ^{10}}{x ^{10} + 1 0 ^{10}}$ is equal to

0

1

10

100

$x \to \infty lim \frac{( x + 1 ) ^{10} + ( x + 2 ) ^{10} + \dots . + ( x + 100 ) ^{10}}{x ^{10} + 1 0 ^{10}}$
$= x \to \infty lim \frac{x ^{10} [ ( 1 + \frac{1}{x} ) ^{10} + ( 1 + \frac{2}{x} ) ^{10} + \dots + ( 1 + \frac{100}{x} ) ^{10} ]}{x ^{10} [ 1 + \frac{1 0 ^{10}}{x ^{10}} ]} = 100.$

Q. x→∞lim​x10+1010(x+1)10+(x+2)10+….+(x+100)10​ is equal to

0

1

10

100

x→∞lim​x10+1010(x+1)10+(x+2)10+….+(x+100)10​ =x→∞lim​x10[1+x101010​]x10[(1+x1​)10+(1+x2​)10+…+(1+x100​)10]​=100.

Q. $x \to \infty lim \frac{( x + 1 ) ^{10} + ( x + 2 ) ^{10} + \dots . + ( x + 100 ) ^{10}}{x ^{10} + 1 0 ^{10}}$ is equal to

$x \to \infty lim \frac{( x + 1 ) ^{10} + ( x + 2 ) ^{10} + \dots . + ( x + 100 ) ^{10}}{x ^{10} + 1 0 ^{10}}$
$= x \to \infty lim \frac{x ^{10} [ ( 1 + \frac{1}{x} ) ^{10} + ( 1 + \frac{2}{x} ) ^{10} + \dots + ( 1 + \frac{100}{x} ) ^{10} ]}{x ^{10} [ 1 + \frac{1 0 ^{10}}{x ^{10}} ]} = 100.$