Limits, Continuity and Differentiability
Limits, Continuity and Differentiability tests evaluation of limits, one-sided behaviour, continuity at a point and derivative rules.
Core rules
- A limit exists only when left-hand and right-hand limits agree.
- For continuity at x=a, f(a) must be defined, the limit must exist, and limit f(x)=f(a).
- Differentiability at a point requires the derivative limit from both sides to exist and agree; differentiability implies continuity, but continuity does not imply differentiability.
- Standard derivatives include d/dx x^n=nx^(n-1), sin x to cos x, e^x to e^x, ln x to 1/x; use product, quotient and chain rules.
Worked example
Let f(x)=(x^2-9)/(x-3) for x≠3 and f(3)=k. Find k for continuity. For x≠3, f(x)=x+3 after factorising. The limit as x→3 is 6. Continuity requires k=6. The value at the point is not obtained by direct substitution into the original quotient.
Traps
- 0/0 is indeterminate, not automatically zero or undefined.
- Check one-sided limits at piecewise breakpoints and absolute values.
- |x| is continuous at 0 but not differentiable there.
- Do not differentiate a function at a point by first simplifying across a breakpoint without checking.
- A derivative may fail to exist where the graph has a corner, cusp or vertical tangent.