Question

Which of the following is correct about the function f(x)=|x+5|?

(a) Left and right limits are equal and hence it is differentiable

(b) Left and right limits are not equal and hence it is differentiable

(c) Left and right limits are equal and hence it is not differentiable

(d) Left and right limits are not equal and hence it is not differentiable

I had been asked this question during an interview.

My doubt stems from Differentiability topic in portion Complex Function Theory of Engineering Mathematics

Answer 1

The correct option is (d) Left and right limits are not equal and hence it is not differentiable

For explanation: f(x)=(x+5), if x>-5 and f(x)=-(x+5), if x≤-5

Since the break is at x=-5, we calculate the limit at this point.

We know that, a function is not differentiable at point x=a, if either \(\lim_{h \to 0}\frac{f(a+h)-f(a)}{h}\)does not exist or is infinity.  We check limits for both the cases of the function.

Left limit: Here, a=-5.

\(\lim_{h \to 0}\frac{f(-5+h)-f(-5)}{h}\)

\(\lim_{h \to 0}\frac{-(-5+h+5)-(-(-5+5))}{h}\)

=-1

Right limit: Here, a=1.

\(\lim_{h \to 0}\frac{f(1+h)-f(1)}{h}\)

\(\lim_{h \to 0}\frac{(1+h+5)-(1+5)}{h}\)

=1

Since the two limits are not equal, the function is not differentiable.