Limits

Definition:

Let f be a function of real variable x. let c, l be two fixed numbers. If f(x) approaches the value L as x approaches a, we say L is the limit of the function f(x) as x tends to a. This is written as

Right hand limit:

Here x>a.

Left hand limit:

Here x

Limit at Infinity:

  We say   if we can make f(x) as close to L as we want by taking x large enough and positive.

Infinite limit:

We say if  we can make f(x) arbitrarily large and positive by taking x sufficiently close to a without letting x=a.

Similarly

we make f(x) arbitrarily large and negative.

Continuous function:

If f(x) is continuous at a then 

Continuous and composition of two functions:

 f(x) is continuous at b and   then

Example:

L’ Hospital’s Rule

a is a number, ∞ or −∞

Example:

Some continuous functions:

• Polynomials for all x.
• Rational function, except for x’s that give division by zero.
• cosx and sinx for all x
• tan(x) and sec(x) provided x ≠ …,
• cot(x) and csc(x) provided x ≠ …,
• e^x for all x .
• In x for x > 0.
•  ⁿ√x   ( n odd) for all x
• ⁿ√x  ( n even)  for all x ≥ 0

Intermediate value theorem:

Suppose that f(x) is continuous on [a, b] and let N be any number between f(a) and f(b). Then there exists a number c such that a < c < b and f(c)=N.

Derivatives:

If y = f(x) then the derivative is defined to be f’(x)=