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.
$$\frac{d}{dx} (x^n) = n x^{n-1}$$
$$\frac{d}{dx} (\sin x) = \cos x$$
$$\frac{d}{dx} (\cos x) = -\sin x$$