# Limits:

As we see in everyday life that everything can’t be exact. The length of an object can’t be a whole number. When we say this rope is of length 5 metre it doesn’t meet with the fact that it is exactly 5 metre. It can be of 4.999999 meters or maybe 5.000001 meters. To deal with such situation limits is introduced in calculus. In this section, we will read limits formula and its properties.

## Left and Right-Hand Limits:

1. Left-Hand Limits:

When the expected value of the function is denoted by the points to the left of a fixed point defines the left-hand limit of the function at that point.

1. Right-Hand Limits:

When the expected value of any function is at right side of the fixed point than we say it as right-hand limit of that point.

## Representation of limit:

A limit can be normally expressed as:
$$\lim _{n\to c}f(n)=L$$

## Properties and algebra of limits:

Let p and q be two functions and a be a value such that $$\displaystyle{\lim_{x \to a}f(x)}$$ and $$\displaystyle{\lim_{x \to a}g(x)}$$ exists at that value:

1. Limit of sum of two functions is sum of the limits of the functions, i.e.,

$$\displaystyle{\lim_{x \to a}[f(x) + g (x)] = \lim_{x \to a}f(x) + \lim_{x \to a}g (x)}$$

1. Limits formula of difference of two functions is difference of the limits of the functions, i.e.,

$$\displaystyle{\lim_{x \to a}[f(x) – g (x)] = \lim_{x \to a}f(x) – \lim_{x \to a}g (x)}$$

1. Limits for any real number k, $$\displaystyle{\lim_{x \to a}[k f(x)] = k \lim_{x \to a}f(x)}$$
2. Limits formula of product of two functions is product of the limits of the functions, i.e.,

$$\displaystyle{\lim_{x \to a}[f(x)\; g(x)] = \lim_{x \to a}f(x) \times \lim_{x \to a}g(x)}$$

1. Limit of quotient of two functions is quotient of the limits of the functions (whenever the denominator is non zero), i.e.,

$$\displaystyle{\lim_{x \to a}\frac{f(x)}{g(x)}} =\frac{\displaystyle{\lim_{x \to a}f(x)}}{\displaystyle{\lim_{x \to a}g(x)}}$$

## Standard Limits:

1. $$\displaystyle{\lim_{x \to a}\frac{x^{n} – a^{n}}{x – a}}$$ = nan-1
2. $$\displaystyle{\lim_{x \to 0}\frac{sin x}{x}}$$ = 1
3. Also, $$\displaystyle{\lim_{x \to 0}\frac{1 – cos x}{x}}$$ = 0

## Limits formula based examples:

1. Find the limit $$\displaystyle{\lim_{x \to 1}\:[x^{3} – x^{2} + 1]}$$.

Solution:
$$\displaystyle{\lim_{x \to 1}}$$ [x3 – x2 + 1] = 13 – 12 + 1 = 1

1. Find the limit $$\displaystyle{\lim_{x \to 1}\frac{x^{2} + 1}{x + 100}}$$.

Solution:
$$\displaystyle{\lim_{x \to 1}\frac{x^{2} + 1}{x + 100}}$$ = $$\frac{1^{2} + 1}{1 + 100}$$ =  $$\frac{2}{101}$$