13. Limits and Derivatives

The biggest risk is not taking any risk. In a world that’s changing really quickly, the only strategy that is guaranteed to fail is not taking risks. – Mark Zuckerburg

The expected value of the function as dictated by the points to the left of a point defines the left hand limit of the function at that point. Similarly the right hand limit.

Limit of a function at a point is the common value of the left and right hand limits, if they coincide.

For a function f and a real number a, \lim_{x \to a} {f(x)} and f(a) may not be same (In fact, one may be defined and not the other one).

For functions f and g the following holds:

\lim_{x \to a} [f(x) \pm g(x)] = \lim_{x \to a}f(x) \pm \lim_{x \to a}g(x)
\lim_{x \to a} [f(x) \times g(x)] = \lim_{x \to a}f(x) \times \lim_{x \to a}g(x)
\lim_{x \to a} \frac{f(x)} {g(x)} = \frac {\lim_{x \to a} f(x)} {\lim_{x \to a} g(x)}

Following are some of the standard limits
\lim_{x \to a} \frac {x^n-a^n}{x-a} = n a^{n-1}
\lim_{x \to 0} \frac {\sin(x)}{x} = 1
\lim_{x \to 0} \frac {1-\cos(x)}{x} = 0

The derivative of a function f at a is defined by

f'(a) =  \lim_{h \to 0} \frac {f(a+h)-f(a)}{h} 

Derivative of a function f at any point x is defined by
f'(x) = \frac{d}{dx} f(x) =  \lim_{h \to 0} \frac {f(x+h)-f(x)}{h} 

For functions u and v the following holds:

(u \pm v)’ = u’ \pm v’
(uv)’ = uv’ + vu’
\left(\frac{u}{v} \right)’=\frac{u’v-uv’}{v^2} provided all are defined.

Following are some of the standard derivatives.
\frac{d}{dx} (x^n) = n x^{n-1}
\frac{d}{dx} (\sin x) = \cos x
\frac{d}{dx} (\cos x) = -\sin x

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