Piecewise Functions - 5

Calculus Level 4

Consider a function $f : \mathbb{R} \rightarrow \mathbb{R}$ such that

$f(x) = \begin{cases} \sin(\pi x) & \text{if } x \in \mathbb{Q} \\ \tan(\pi \sqrt{|x|}) & \text{if }x \not \in \mathbb{Q}\end{cases}$

Find the sum of all positive integers $N < 100$ , such that $\displaystyle\lim _{x \rightarrow N} f(x)$ exists.

This problem is part of the set - Piecewise-defined Functions

1 solution

Pranshu Gaba
Oct 26, 2014

For a function $f$ of the form

$f(x) = \begin{cases} g(x) & \text{if } x \in \mathbb{Q} \\ h(x) & \text{if } x \not \in \mathbb{Q} \end{cases}$

$\displaystyle\lim_{x\rightarrow N} f(x)$ will exist iff $g(N) = h(N)$ .

So, we just need to find integer values of $N$ such that $\sin (\pi N )= \tan (\pi \sqrt{N})$

We know that $\sin (\pi N)$ will be $0$ for all integers $N$ . So, $\tan(\pi \sqrt{N})$ must also be $0$ .

Therefore $\sqrt{N}$ must be an integer, so $N$ must be a perfect square.

The possible values of $N$ are $1^2, 2^2, 3^2, 4^2, 5^2, 6^2, 7^2, 8^2$ and $9^2$ .

The Answer = $1 + 4 + 9 + 16 + 25 + 36 + 49 + 64 + 81 = \frac{9 \times 10 \times 19}{6} = \boxed{285}$