Limits is all about the output function, if x is really close to a limiting number, infinity or zero.
Limits are used when when we want to evaluate a function at a particular value, but can’t substitute the value in straight because the formula “blows up”. You have to do something to the formula first.

There are few types of limits….

Type 1: In first type we take an equation with a variable x, and see what number it approaches at it reaches infinity. A simple one is 1/x. As x gets closer to infinity, it gets closer and closer to a value of zero, but it will never actually be zero.

Type 2: The other type of limit is the limit of a sum or integral of an equation as x goes from a to b where either a or b or both are infinity. Obviously something like the limit of the sum of 1 + x from 0 to infinity does not have a limit as 1+x keeps getting larger. But some equations do decrease as x gets higher than value and some of those would decrease enough that the upper limit of that sum will approach a value, but never reach it.

Basic Facts

When does a formula “Blow Up”?

X/0, when you have to divide a value from zero.
0/0, when you have to divide zero by zero.

Sides of Limits

There are two sides of limits known as left-hand limits and right-hand limits.

When does a limit not exist?

• When the left-and right-hand limits aren’t equal (resulting in a discontinuity in the function).
• When  a function increases or decreases infinitely (“without bound”) as it approaches a given x-value.
• In the cases of infinite oscillation when approaching a fixed point

Approaching Infinity

1/∞ is undefined. But we can still approach to a value.

Evaluating methods for Limits (E-value-ating)

1. Substitution Method
2. Factorizen Method
3. Conjugate Method
4. Limits of Rational Functions
5. L’Hôpital’s Rule
6. Formal Method

Substitution Method – Limits Evaluating

In substitution method, we put the value of the limit to the equation and check whether it gets a valid answer.

Factorizen Method – Limits Evaluating

In factoring method, we solve the equation by factoring  the numerator or the denominator, or the both. 

Conjugate Method – Limits Evaluating

In Conjugate method, we multiply numerator and the denominator of a fraction by one of it’s conjugates. 

Limits of Rational Functions -Limits Evaluating

A rational function is a type of function, which is the both the numerator and the denominator are polynomials.
When finding Limits of Rational Functions if the limit goes to infinity, we can consider the overall degree of the function to apply method of Limits to Infinity.