3. Limits of polynomial, rational, trigonometric, exponential and logarithmic functions

Limits of polynomial, rational, trigonometric, exponential and logarithmic functions

Limits of polynomials and rational functions- A function f is said to be a polynomial function of degree n f(x) = a⁰ + a¹x + a²x² +. . . + aⁿxⁿ, where a_i ‘s are real numbers such that an ¹ 0 for some natural number n.

Limits of Rational Functions:

By evaluating the function at 2, we got 0/0.

So, let us factorise the functions to cancel the factors if possible.

4x² – 6x – 18 = 4x² – 12x + 6x – 18

= 4x(x – 3) + 6(x – 3)

= (x – 3)(4x + 6)

And x² – 9 = x² – 3² = (x – 3)(x + 3)

Thus,

Limits of Trigonometric Functions:

Theorem: Let f and g be two real valued functions with the same domain such that

f (x) £ g( x) for all x in the domain of definition, For some a, if both limx®a f(x) and limx®a g(x) exist, then limx®a f(x) £ limx®a g(x).

Sandwich Theorem/Squeeze Principle :

If f, g and h are three functions such that f(x) < g(x) < h(x) for all x in some interval containing the point x = a, and If

Some standards limits formulae are:

Limits of polynomials and rational functions:  A function f is said to be a polynomial function of degree n f(x) = a0 + a1x + a2x2 +. . . + anxn, where ais are real numbers such that an ¹ 0 for some natural number n.

We know that limx->a  x = a. Hence