1# Differential calculus23## Limits45$$\lim_{x \rightarrow a}f(x)$$67$L^-$ - limit from below89$L^+$ - limit from above1011$\lim_{x \to a} f(x)$ - limit of a point1213- Limit exists if $L^-=L^+$14- If limit exists, point does not.1516Limits can be solved using normal techniques (if div 0, factorise)1718## Limit theorems19201. For constant function $f(x)=k$, $\lim_{x \rightarrow a} f(x) = k$212. $\lim_{x \rightarrow a} (f(x) \pm g(x)) = F \pm G$223. $\lim_{x \rightarrow a} (f(x) \times g(x)) = F \times G$234. ${\lim_{x \rightarrow a} {f(x) \over g(x)}} = {F \over G}, G \ne 0$2425Corollary: $\lim_{x \rightarrow a} c \times f(x)=cF$ where $c=$ constant2627## Solving limits for $x\rightarrow\infty$2829Factorise so that all values of $x$ are in denominators.3031e.g.3233$$\lim_{x \rightarrow \infty}{{2x+3} \over {x-2}}={{2+{3 \over x}} \over {1-{2 \over x}}}={2 \over 1} = 2$$34353637## Continuous functions3839A function is continuous if $L^-=L^+=f(x)$ for all values of $x$.