# Limit to Function - ISI UG 2019 Subj Problem 2

## Understand the problem

[/et_pb_text][et_pb_text _builder_version="3.27.4" text_font="Raleway||||||||" background_color="#f4f4f4" custom_margin="10px||10px" custom_padding="10px|20px|10px|20px" box_shadow_style="preset2"]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.

I.S.I. (Indian Statistical Institute, B.Stat, B.Math) Entrance. Subjective Problem 2 from 2019
[/et_pb_accordion_item][et_pb_accordion_item title="Topic" _builder_version="3.22.4" open="off"]Calculus

[/et_pb_accordion_item][et_pb_accordion_item title="Difficulty Level" _builder_version="3.22.4" open="off"]6 out of 10

[/et_pb_accordion_item][et_pb_accordion_item title="Suggested Book" _builder_version="4.3.4" hover_enabled="0" open="on"]

[/et_pb_text][et_pb_tabs active_tab_background_color="#0c71c3" inactive_tab_background_color="#000000" _builder_version="3.22.4" tab_text_color="#ffffff" tab_font="||||||||" background_color="#ffffff"][et_pb_tab title="Hint 0" _builder_version="3.22.4"]

Do you really need a hint? Try it first!

[/et_pb_tab][et_pb_tab title="Hint 1" _builder_version="3.22.4"]

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.

[/et_pb_tab][et_pb_tab title="Hint 2" _builder_version="3.22.4"]

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

[/et_pb_tab][et_pb_tab title="Hint 3" _builder_version="3.22.4"]

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

[/et_pb_tab][et_pb_tab title="Hint 4" _builder_version="3.22.4"]

Prove that  $f(x)$ = $\left\{ \begin{array}{ll} 0 & 0 < x < \frac{1}{2} \\ \frac{1}{\sqrt{e}} & x = \frac{1}{2} \\ 1 & x > \frac{1}{2} \\ \end{array} \right.$

## Connected Program at Cheenta

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.