Continuity at a means that the function value agrees with the common finite limit from both sides. Differentiability means that the limiting tangent gradient exists and is finite. Differentiability implies continuity; continuity alone does not imply differentiability. The graph y=|x| is continuous at zero but has left gradient -1 and right gradient 1, so it is not differentiable there. These ideas are informal; the current guide does not require examination tests of continuity or differentiability. **Behaviour** | **Example** Converges to a finite limit | lim_x to infinity 2x+1x+3=2 Grows without bound | x2 to +infinity as x to infinity No common two-sided limit | 1x to +infinity as x 0+, but 1x to -infinity as x 0- Divergence need not mean unbounded growth: as x to infinity, sin x remains between -1 and 1 but does not approach one value. The symbol infinity describes unbounded growth; it is not a real number that can be substituted into an expression. A numerical table should approach from both admissible sides. One calculator value close to a point does not establish a limit.

Continuity at \(a\) means that the function value agrees with the common finite limit from both sides. Differentiability means that the limiting tangent gradient exists and is finite. Differentiability implies continuity; continuity alone does not imply differentiability.

The graph \(y=|x|\) is continuous at zero but has left gradient \(-1\) and right gradient \(1\), so it is not differentiable there. These ideas are informal; the current guide does not require examination tests of continuity or differentiability.

Divergence need not mean unbounded growth: as \(x\to\infty\), \(\sin x\) remains between \(-1\) and \(1\) but does not approach one value. The symbol \(\infty\) describes unbounded growth; it is not a real number that can be substituted into an expression. A numerical table should approach from both admissible sides. One calculator value close to a point does not establish a limit.

Limits and Continuity

International Baccalaureate Mathematics: Analysis and Approaches Higher Level

Behaviour	Example
Converges to a finite limit	\(\lim_{x\to\infty}\dfrac{2x+1}{x+3}=2\)
Grows without bound	\(x^2\to+\infty\) as \(x\to\infty\)
No common two-sided limit	\(\dfrac1x\to+\infty\) as \(x\to0^+\), but \(\dfrac1x\to-\infty\) as \(x\to0^-\)

Continuity, Differentiability and Limits

Continuity, Differentiability and Limits