lim _{x} {rightarrow infty} frac{(2 x+1)^{50} | vedclass

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Question

(B) We are given the limit $L = \lim _{x \rightarrow \infty} \frac{\sum_{k=1}^{100} (2x+k)^{50}}{(2x)^{50} + 10^{50}}$.Divide the numerator and denomi

Vedclass · Accepted Answer

$100$

Vedclass · Answer

(B) We are given the limit $L = \lim _{x \rightarrow \infty} \frac{\sum_{k=1}^{100} (2x+k)^{50}}{(2x)^{50} + 10^{50}}$.Divide the numerator and denominator by $(2x)^{50}$:$L = \lim _{x}$ ${\rightarrow \infty} \frac{\sum_{k=1}^{100} \left(\frac{2x+k}{2x}\right)^{50}}{1 + \frac{10^{50}}{(2x)^{50}}}$.As $x \rightarrow \infty$,the term $\frac{k}{2x} \rightarrow 0$ for each $k \in \{1, 2, \dots, 100\}$.Thus,$\left(1 + \frac{k}{2x}\right)^{50} \rightarrow 1^{50} = 1$.The numerator becomes $\sum_{k=1}^{100} 1 = 100$.The denominator becomes $1 + 0 = 1$.Therefore,$L = \frac{100}{1} = 100$.

$\lim _{x}$ ${\rightarrow \infty} \frac{(2 x+1)^{50}+(2 x+2)^{50}+(2 x+3)^{50}+\cdots \cdots+(2 x+100)^{50}}{(2 x)^{50}+(10)^{50}} = \dots$

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