lim_xtooo(1+b/x)^(x/a)=?

প্রশ্ন: \( \lim_{x \to \infty} \left(1 + \frac{b}{x}\right)^{\frac{x}{a}} = ? \) 🤔

সমাধান:

আমরা জানি, \( \lim_{x \to \infty} \left(1 + \frac{c}{x}\right)^{x} = e^c \) 🤩।

এখন, প্রদত্ত রাশিমালা:

\( \lim_{x \to \infty} \left(1 + \frac{b}{x}\right)^{\frac{x}{a}} \)

= \( \lim_{x \to \infty} \left[ \left(1 + \frac{b}{x}\right)^{x} \right]^{\frac{1}{a}} \) 💪

যেহেতু, \( \lim_{x \to \infty} \left(1 + \frac{b}{x}\right)^{x} = e^b \), তাই আমরা লিখতে পারি:

= \( \left[ \lim_{x \to \infty} \left(1 + \frac{b}{x}\right)^{x} \right]^{\frac{1}{a}} \)

= \( \left(e^b\right)^{\frac{1}{a}} \)

= \( e^{\frac{b}{a}} \) 🎉

অতএব, \( \lim_{x \to \infty} \left(1 + \frac{b}{x}\right)^{\frac{x}{a}} = e^{\frac{b}{a}} \) 🥳