Question

Let f(x) = \(\frac{sin(x)}{x – 54}\), then the value of f^(100)(54) is given by?

(a) Undefined

(b) 100

(c) 10

(d) 0

I have been asked this question during an interview for a job.

This interesting question is from The nth Derivative of Some Elementary Functions topic in section Differential Calculus of Engineering Mathematics

Answer 1

Correct option is (a) Undefined

To explain: The key here is to expand the numerator into a taylor series centered at 54

Doing this gives us the following

Sin(x)=Sin(54)+\(\frac{(x-54)\times Cos(54)}{1!}-\frac{(x-54)^2\times Sin(54)}{2!}…\infty \)

Now the function transforms into f(x)=\(\frac{Sin(54)}{(x-54)}+\frac{Cos(54)}{1!}-\frac{(x-54)\times Sin(54)}{2!}…\infty \)

Observe the first term in the infinite series there is always (x – 54) in the denominator which goes to 0 when we substitute x = 54

Every derivative also will have this term

Hence any derivative of the given function goes to ∞ as x = 54

Hence, the answer is Undefined.