# Introduction to Limits

Let f(x) be a function at an interval that contains x = c (except possibly at c) and let 'L' be any real number such that

$\lim_{x->c}f(x)=L$
It means that for each $\varepsilon$ >0, there exists $\delta$ >0 such that if

0 < |x - c| <$\delta$ then

|f(x) - L | < $\varepsilon$
There are different types of ways to find the limit.
1) Finding limits Graphically.
2) Finding limits numerically.
3) Finding limits analytically.
Here apart from the above methods, you will learn
1) different ways that a limit can fail to exists.
2) study and use a formal definition of limit.

## Introduction of Limit in the Real World

Example 1 : Growth and decay of bacteria to the population of squirrels. When we want to know what happens in the long time, there we use the limit.
Example 2 : Many buses are moving on the road. Traffic flow limits and here the limits are nothing but the bus stops. If we remove one limit(bus stop), the traffic or flow will increase till you will get the next limit( bus stop).
Example 3: In whatsapp there is a limit of 5 for forwarding any message or video.
Example 4: When any guests are coming to our place for the dinner then we have to estimate the limit to how much one can eat.

Limits :

• finding limits numerically
• finding limits graphically
• limit that fail to exist
• $\varepsilon- \delta$ definition of limit
• properties of limits
• evaluate limits by direct substitution
• limits of trigonometric functions
• finding limits by factorization method
• finding limits by rationalization method
• Calculus squeeze theorem
• Continuity of a function
• Greatest integer function
• Intermediate value theorem
• evaluating limit by using standard limits
• finding algebraic limits at infinity