Limits

The concept of limits is essential to calculus. The limit is the basis for all calculus problems. A good understanding of limits will help explain many theories in calculus.

Consider a function f defined for values of x, as x gets close to a number a, not necessarily true for x = a. If the value of f(x) approaches a number b as x approaches a, then the limit of f(x) as x approaches a is equal to b, denoted as :

lim_x®af(x) = b

ex.
Find the limit of f(x) = 3x + 2 as x approaches 3.

It is said that as x approaches 3, 3x approaches 9, and 3x + 2 approaches 11. Thus;

lim_x®3 3x + 2 = 11

Find the limit of f(x) = 1/ x - 3 as x approaches 3.

It is said that as x approaches 3, x - 3 approaches 0, and 1/ x - 3 approaches 1/ 0 which is undefined. Thus;

lim_x®3 1/x - 3 = undefined

In limits, the only thing that matters is how a function is defined near the point a.

Elementary Limits:

lim_x®af(x) = f(x)

This limit represents a horizontal line, which says that as x approaches a,

f(x) = f(x).

lim_x®af(x) = a

This limit says that as x approaches a, f(x) also approaches a.

Addition, Subtraction, Multiplication and Division of Limits:

If lim_x®af(x) = S and lim_x®ag(x) = T then,

lim_x®a [f(x) - g(x)] = S + T

lim_x®a [f(x) - g(x)] = S - T

lim_x®a [f(x) - g(x)] = S · T

Limits Of Polynomials and Rational Functions:

If P(x) and Q(x) are polynomials, then:

lim_x®a P(x) = P(a)

and

Limits can be approached from the negative ( or left ) or the positive ( or right ) side of a number denoted as:

lim_x®a-f(x) = b or lim_x®a+f(x) = b

If lim_x®a-f(x) ¹ lim_x®a+f(x) , then

lim_x®af(x) does not exist

If lim_x®a-f(x) = lim_x®a+f(x) = b , then

lim_x®af(x) = b

Infinite Limits:

If the value of f(x) gets larger and larger without bound as x approaches a, then :

lim_x®a-f(x) = +

Similarly; If the value of f(x) gets smaller and smaller without bound as x approaches a, then :

lim_x®af(x) = -¥

Limits at Infinity:

Consider a function f defined for large positive ( or negative ) values of x, as x increases indefinitely in the positive ( or negative ) direction. If the value of f(x) approaches a number b as x increases (or decreases ) indefinitely , then the limit of f(x) as x increases (or decreases ) indefinitely is equal to b, denoted as :

lim_x®+¥ f(x) = b or lim_x®-¥ f(x) = b

Continuity:

A function f is continuos at x = a if f is defined at x = a and either ; f is not defined anywhere near a; or f is defined arbitrarily near x = a and lim_x®af(x)=f(a).

Conversely, A function f is discontinuous at x = a if f is defined at x = a and f is not continuos at x = a.

Addition, Subtraction, Multiplication and Division of Continuity:

If f and g are continuos functions at x = a then; f + g, f - g, f · g and f/g; where g(a) ¹ 0; are also continuous at x = a.

If the function f is a polynomial or a rational function then f is continuos wherever it is defined.

Given f(g(x)), where g is continuos at x = a , and f is continuos at x = g(a) then f(g(x)) is continuos at x = a.

More Graphing

Intercepts and Asymptotes:

Given an equation of the form:

f(x) = P(x) / Q(x) in reduced form;

If (x - a)ⁿ (where n is a positive integer) is a factor of either P(x) or Q(x) and if (x - a)ⁿ⁺¹ is a factor of neither , then

a. The graph crosses the x axis at x = a if and only if n is odd.
b. The graph stays on the same side of the x axis at x = a if and only if n is even.

Vertical Asymptotes:

The line x = a is a vertical asymptote of the function f if at least one of the following statements are true:

Horizontal Asymptotes:

The line y = b is a horizontal asymptote of the function f if at least one of the statements is true:

lim_x®+¥ f(x) = b or lim_x®-¥ f(x) = b

Slant Asymptotes:

The line y = ax + b is a slant asymptote of a function f if:

lim_{x®± ¥} [ f(x) - (ax + b)] = 0