IB Maths AA Topic 5 Notes

Limits

Calculus studies change. In this section, the key ideas are limits, derivatives as gradients and rates of change, increasing and decreasing functions, and the derivative rule for powers and polynomials with integer exponents.

$\lim_{x\to a}f(x)=A$

A limit describes the value a function approaches as the input approaches a particular value. If $f(x)$ gets closer and closer to $A$ as $x$ gets closer to $a$, then we write $\lim_{x\to a}f(x)=A$.

At this level, limits are introduced informally and are usually estimated from tables or graphs rather than by formal analytic methods.

Supposing $f(x)=\frac{x^2-9}{x-3}$, estimate $\lim_{x\to3}f(x)$. 

Note that $f(x)$ is undefined at $x=3$, but values close to $3$ can still show what the function approaches.

• When $x=2.9$, $f(x)=5.9$
• When $x=2.99$, $f(x)=5.99$
• When $x=3.01$, $f(x)=6.01$
• When $x=3.1$, $f(x)=6.1$

The outputs approach $6$, so $\lim_{x\to3}\frac{x^2-9}{x-3}=6$.

If a graph approaches the same $y$-value from both sides of $x=a$, then that common $y$-value is the limit.

The actual value of the function at $x=a$ does not matter for the limit. A graph may have a hole at that point and still have a limit there.

This sketch should show a removable discontinuity: the curve approaches the same $y$-value from both sides, but there is a hole at $x=a$.