- Difficulty
- Challenging
- Suggested time
- 5 min
- Marks
- 3
Exponential and trigonometric functions practice question 19
Let . Which statement about at is true?
Continuity and differentiability are separate questions; settle each one. What formula does follow on each side of 0?
Write without the absolute value: for and for .
Compute the one-sided limits of for and for , using .
Theory UBC Math 100 §4.2CLP-1 §2.8Key results
Worked solution
Strategy
Split at 0 using the definition of |x|, check continuity, then compare the one-sided limits of the difference quotient.
Working
For negative x, |x| = −x and sin(−x) = −sin x.
Both sides tend to f(0), so f is continuous at 0.
The standard limit sin h / h → 1, taken from each side.
The graph has a corner at 0, like |x| itself.
Answer
The answer is f is continuous at 0 but not differentiable there.
Check
Near 0, sin|x| ≈ |x|: at x = ±0.01 both are about 0.01, so the graph has the corner of |x|, with slopes −1 and 1.
Where this goes wrong
Concluding f′(0) = 0 because f is even, which assumes the derivative exists. A corner is continuous, and here the one-sided slopes, −1 and 1, do not agree.
Leans on
Differentiability, from The definition of the derivative
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Let . Which statement about at is correct?Saved in this browser only.