1# Differential calculus 2 3## Limits 4 5$$\lim_{x \rightarrow a}f(x)$$ 6 7$L^-$ - limit from below 8 9$L^+$ - limit from above 10 11$\lim_{x \to a} f(x)$ - limit of a point 12 13- Limit exists if $L^-=L^+$ 14- If limit exists, point does not. 15 16Limits can be solved using normal techniques (if div 0, factorise) 17 18## Limit theorems 19 201. 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. ${\lim_{x \rightarrow a} {f(x) \over g(x)}} = {F \over G}, G \ne 0$ 24 25Corollary: $\lim_{x \rightarrow a} c \times f(x)=cF$ where $c=$ constant 26 27## Solving limits for $x\rightarrow\infty$ 28 29Factorise so that all values of $x$ are in denominators. 30 31e.g. 32 33$$\lim_{x \rightarrow \infty}{{2x+3} \over {x-2}}={{2+{3 \over x}} \over {1-{2 \over x}}}={2 \over 1} = 2$$ 34 35 36## Continuous functions 37 38A function is continuous if $L^-=L^+=f(x)$ for all values of $x$. 39 40## Gradients of secants and tangents 41 42Secant (chord) - line joining two points on curve 43 44Tangent - line that intersects curve at one point 45 46given $P(x,y) \quad Q(x+\delta x, y + \delta y)$: 47gradient of chord joining $P$ and $Q$ is ${m_{PQ}}={\operatorname{rise} \over \operatorname{run}} = {\delta y \over \delta x}$ 48 49As $Q \rightarrow P, \delta x \rightarrow 0$. Chord becomes tangent (two infinitesimal points are equal). 50 51Can also be used with functions, where $h=\delta x$. 52 53## First principles derivative 54 55$$\lim_{\delta x \rightarrow 0}{\delta y \over \delta x}={dy \over dx} = f^\prime(x)$$ 56 57$$m_{\operatorname{tangent}}=\lim_{h \rightarrow 0}f^\prime(x)$$ 58 59 60 61$$m_{\operatorname{chord PQ}}=f^\prime(x)$$ 62 63first principles derivative: 64$${m_{\operatorname{tangent at P}} =\lim_{h \rigzhtarrow 0}}{{f(x+h)-f(x)}\over h}$$ 65 66 67 68 69## Euler's number as a limit 70 71$$\lim_{h \rightarrow 0} {{e^h-1} \over h}=1$$