Understand the problem

Let \( f\) : \( (0,1) \rightarrow \mathbb{R} \) be defined by

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

(a) Show that \(f\) has exactly one point of discontinuity.

(b) Evaluate \(f\) at its point of discontinuity.

Source of the problem

I.S.I. (Indian Statistical Institute, B.Stat, B.Math) Entrance. Subjective Problem 2 from 2019

Topic

Calculus

Difficulty Level

6 out of 10

Suggested Book

Calculus in one variable by I.A.Maron

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

\(f(x)\) =

\[\left\{ \begin{array}{ll}
0 & 0 < x < \frac{1}{2} \\
\frac{1}{\sqrt{e}} & x = \frac{1}{2} \\
1 & x > \frac{1}{2} \\ \end{array} \right. \]

Watch the video ( Coming Soon … )

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.

Similar Problems

An inequality with many unknowns

Understand the problemLet be positive real numbers such that . Prove thatSingapore Team Selection Test 2008InequalitiesMediumInequalities by BJ VenkatachalaStart with hintsDo you really need a hint? Try it first!Use the method of contradiction.Suppose that $latex...

Looks can be deceiving

Understand the problemFind all non-zero real numbers which satisfy the system of equations:Indian National Mathematical Olympiad 2010AlgebraMediumAn Excursion in MathematicsStart with hintsDo you really need a hint? Try it first!When a polynomial equation looks...

IMO, 2019 Problem 1 – Cauchyish Functional Equation

This problem is a patient and intricate and simple application of Functional Equation with beautiful equations to be played aroun with.

A sequence of natural numbers and a recurrence relation

Understand the problemDefine a sequence by , andfor For every and prove that divides. Suppose divides for some natural numbers and . Prove that divides Indian National Mathematical Olympiad 2010 Number Theory Medium Problem Solving Strategies by Arthur Engel...

Linear recurrences

Linear difference equationsA linear difference equation is a recurrence relation of the form $latex y_{t+n}=a_1y_{t+n-1}+a_2y_{t+n-2}+\cdots +a_ny_t+b$. If $latex b=0$, then it is called homogeneous. In this article, we shall also assume $latex t=0$ for...

Functional Equation PRMO 2012 Problem 16

A beautiful functional equation problem from PRMO (Pre Regional Math Olympiad) 2012 (Problem 16). Use sequential hints and a video discussion to try the problem

2013 AMC 10B – Problem 5 Maximizing the Difference:

This is based on simple ineqaulities on real numbers.

An inductive inequality

Understand the problemGiven and for all , show that Singapore Mathematical Olympiad 2010 Inequalities Easy Inequalities by BJ Venkatachala Start with hintsDo you really need a hint? Try it first!Use induction. Given the inequality for $latex n=k$, the inequality...

A search for perfect squares

Understand the problemDetermine all pairs of positive integers for which is a perfect square.Indian National Mathematical Olympiad 1992 Number Theory Easy An Excursion in Mathematics Start with hintsDo you really need a hint? Try it first!First consider $latex n=0$....

INMO 1996 Problem 1

Understand the problema) Given any positive integer , show that there exist distint positive integers and such that divides for ; b) If for some positive integers and , divides for all positive integers , prove that .Indian National Mathematical Olympiad...