Limits

The value of a function in the neighbourhood of a point, where the function is discontinuous is called the limiting value.

Limits
There are three ways in which a function can approach the limiting value

1) LHL and RHL approach the same value . Limit exists.

2) LHL and RHL approach two different limiting values . Limit does not exist, but we can find the individual limiting values for both.

3) The Graph has a weird behavior or trajectory.

The limitting Value can be infinity e.g.  1/|x|.

Reasons for Non - Existence of Limits
1) Function is not defined in the neighbourhood of x . (even if one of the limits is not defined )

2) LHL and RHL approach different values.

3) The Graph has a weird behaviour.

L-Hospital's Rule
If while, evaluating the limits , we get infinity or an undefined value , then we can differentiate the numerator and denominator separately , till the infinity factor is removed.

For MCQs with limits of the form 0/0, L-Hospital's Rule is crucial.

Evaluation of Limits
Types:-

1) Algebraic Evaluation

2) Trigonometric Evaluation

3) Logarithmic Evaluation

4) Exponential Evaluation

5) Miscellaneous

Hints
1) For evaluating RHL or LHL with a variable a, put x tends to (a + h) or (a - h). 2) While dealing with trigonometric functions , if x does not tend to 0 , use a variable h such that h tends to 0.