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MHTCET 26 Apr 2025 Shift 1 Paper

Question : 140 of 150

Marks: +1, -0

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

Solution:

Concept:

Divide numerator and denominator by the highest power,

(2x)^{50}

, then apply the limit term by term.

Explanation:

The given limit is:

\lim_{x \to \infty} \frac{\sum_{k=1}^{100} (2x + k)^{50}}{(2x)^{50} + (10)^{50}}

Divide both numerator and denominator by

(2x)^{50}

= \lim_{x \to \infty} \frac{\sum_{k=1}^{100} \left(1 + \frac{k}{2x}\right)^{50}}{1 + \left(\frac{10}{2x}\right)^{50}}

x \to \infty

, we have

\frac{k}{2x} \to 0

and

\frac{10}{2x} \to 0

So each term

\left(1 + \frac{k}{2x}\right)^{50} \to 1^{50} = 1

There are 100 such terms in the numerator, giving

100 \times 1 = 100

The denominator tends to

1 + 0 = 1

Therefore, the limit is

\frac{100}{1} = 100

Answer:

100

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