# 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:**

**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.

**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:

\({\displaystyle \lim _{n\to c}f(n)=L}\)

**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:

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

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

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

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

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

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

- 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:**

- \(\displaystyle{\lim_{x \to a}\frac{x^{n} – a^{n}}{x – a}}\) = na
^{n-1} - \(\displaystyle{\lim_{x \to 0}\frac{sin x}{x}}\) = 1
- Also, \(\displaystyle{\lim_{x \to 0}\frac{1 – cos x}{x}}\) = 0

**Limits formula based examples:**

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

**Solution:**

\(\displaystyle{\lim_{x \to 1}}\) [x^{3} – x^{2} + 1] = 1^{3} – 1^{2 }+ 1 = 1

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

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

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