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1# Differential calculus
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3## Limits
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5$$\lim_{x \rightarrow a}f(x)$$
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7$L^-$ - limit from below
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9$L^+$ - limit from above
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11$\lim_{x \to a} f(x)$ - limit of a point
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13- Limit exists if $L^-=L^+$
14- If limit exists, point does not.
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16Limits can be solved using normal techniques (if div 0, factorise)
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18## Limit theorems
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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. ${\lim_{x \rightarrow a} {f(x) \over g(x)}} = {F \over G}, G \ne 0$
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25Corollary: $\lim_{x \rightarrow a} c \times f(x)=cF$ where $c=$ constant
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27## Solving limits for $x\rightarrow\infty$
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29Factorise so that all values of $x$ are in denominators.
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31e.g.
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33$$\lim_{x \rightarrow \infty}{{2x+3} \over {x-2}}={{2+{3 \over x}} \over {1-{2 \over x}}}={2 \over 1} = 2$$
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35
36
37## Continuous functions
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39A function is continuous if $L^-=L^+=f(x)$ for all values of $x$.