## Understand the problem

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## Start with hints

Do you really need a hint? Try it first!

Try to determine the **Form of the Limit**. Show that the limit is of the form \(1^\infty\). Hence, try to find the **Functional Form**.

\( f(x) = \lim_{n\to\infty} (1 + (cos(\frac{1}{n^x}) – 1))^n = e^{(\lim_{n\to\infty}(cos(\frac{1}{n^x}) – 1).n} \).

\({\lim_{n\to\infty}(cos(\frac{1}{n^x}) – 1).n = -\frac{1}{2}\lim_{n\to\infty}\frac{(sin^2(\frac{1}{2n^x}))}{(\frac{1}{2n^x})^2}.n^{(1-2x)} } \)

Prove that \(f(x)\) = \[\left\{ \begin{array}{ll}

0 & 0 < x < \frac{1}{2} \\

\frac{1}{\sqrt{e}} & x = \frac{1}{2} \\

1 & x > \frac{1}{2} \\ \end{array} \right. \]

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## Connected Program at Cheenta

# I.S.I. & C.M.I. Entrance Program

Indian Statistical Institute and Chennai Mathematical Institute offer challenging bachelor’s program for gifted students. These courses are B.Stat and B.Math program in I.S.I., B.Sc. Math in C.M.I.

The entrances to these programs are far more challenging than usual engineering entrances. Cheenta offers an intense, problem-driven program for these two entrances.