Can you spell the name of this function?

Calculus Level 5

Let $\omega(x) = \displaystyle \sum_{n=0}^{\infty} \dfrac{1}{2^{n}} \sin\left(2^{n}x \right)$ .

Evaluate $\displaystyle \lim_{h\rightarrow 0} \dfrac{\omega(x+h)-\omega(x)}{h}$ at $x=2016^{\circ}$ .

If the answer cannot be inputted into the solution box, select the correct option below and input the number of the option that best fits the correct answer:

Limit DNE, by approaching $+ \infty$ on both sides of 2016: Input 1234
Limit DNE, by approaching $- \infty$ on both sides of 2016: Input 4321
Limit DNE, by not approaching the same value from left and from right of 2016, but not approaching $\pm \infty$ on either side: Input 1324
Limit DNE, by approaching $-\infty$ from the left of 2016, and $+ \infty$ from the right of 2016: Input 1243
Limit DNE, by approaching $+\infty$ from the left of 2016, and $- \infty$ from the right of 2016: Input 3241

The answer is 1324.

1 solution

Hobart Pao
Jan 8, 2016

This is a non rigorous solution. The function given is an example of Weierstrass Function, which is nowhere differentiable though it is everywhere continuous. If you graph the function, you'll notice that if you keep zooming in anywhere, there will be an infinite amount of bumps, but you're not going to have cusps. Thus, the answer is that the limit does not exist (because the derivative is a limit) and it doesn't matter where, because the limit does not exist anywhere, and it is by not having the same slope, approaching from either side of 2016, but not approaching positive or negative infinity.

Can you spell the name of this function?

The answer is 1324.

1 solution

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