Evaluation of Algebraic limits

Direct substitution method
Factorisation method
Rationalization method
Using standard formulas
Method for limits when x→∞
Evaluation of Trigonometric limits
Evaluation of Exponential and Logarithmic Limits
Evaluation of Exponential Limits of the Form 1ˉ

Algebra of Limits

Algebra of Limits

1. lim x→a [f(x) ± g(x)] = lim x→a f(x) ± lim x→ag(x)

2. lim x→a [k.f(x)] = k lim x→af(x)

3. lim x→a[f(x).g(x)] = [lim x→a f(x)][ lim x→a g(x)]

4. lim x→a [f(x)/g(x)] = [lim x→a f(x)]/[ lim x→a g(x)] provided lim x→a g(x) ≠ 0.

5. If f(x) is l.t. g(x) then lim x→a f(x) ≤ lim x→a g(x)

Right Hand and Left Hand Limt

Right Hand and Left Hand Limit

The statement x→aˉ means that x is tending to a from the left hand side.

The statement x→a⁺ means that x is tending to a from the right hand side.

Steps to find left hand limit

Put x = a - h and replace x→aˉ by h→0. Find limit of f(a-h) as h→0

Steps to find right hand limit

Put x = a + h and replace x→a⁺ by h→0. Find limit of f(a+h) as h→0

Limits

Definition

Let f(x) be a function of x. If for every positive number ε ( tends to 0),

then there exist a positive number δ such that whenever

0 < | x - a |< δ => | f(x) - l | < ε ,

then we can say , f(x) tends to limit l as x tends to a

and we can write

lim (x→a) f(x) = l.