Mathematics

Mathematics

Problem

Let f(x)=∣x∣f(x)=|x|. (a) Evaluate lim⁡h→0−f(0+h)−f(0)handlim⁡h→0+f(0+h)−f(0)h.\lim_{h\to 0^-}\frac{f(0+h)-f(0)}{h}\quad\text{and}\quad \lim_{h\to 0^+}\frac{f(0+h)-f(0)}{h}. (b) Based on these limits, determine whether ff is differentiable at x=0x=0.
Intuitive explanation

The derivative measures how a function's output changes relative to its input near a point—but only if this rate of change is the same from both directions. Here, you're checking the left and right limits of the difference quotient for ∣x∣|x| at its "corner" point x=0x=0. If these one-sided limits disagree, the function has a sharp turn and isn't differentiable there; if they match, a derivative exists.

Intuitive explanation

The derivative measures how a function's output changes relative to its input near a point—but only if this rate of change is the same from both directions. Here, you're checking the left and right limits of the difference quotient for ∣x∣|x| at its "corner" point x=0x=0. If these one-sided limits disagree, the function has a sharp turn and isn't differentiable there; if they match, a derivative exists.

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