Calculus Made Easy

Now that we have understood what the limit of a function is, we must now know the various properties surrounding the limits of a function.

There are 7 main properties of limits that we must know in calculus and vectors.

lim k = k, for all constants of k
x->a

lim x = a
x->a

lim [f(x) +/- g(x)] = lim f(x) +/- lim g(x)
x->a x->a x->a

lim [cf(x)] = c[lim f(x)], for all constants of c
x->a x->a

lim [f(x)g(x)] = [lim f(x)] [lim g(x)]
x->a x->a x->a

lim f(x) = lim f(x), given that lim g(x) does not equal to 0
x->a g(x) x->a x->a
lim g(x)
x->a
(The property above this statement is meant to represent what to do when two limits are being divided with each other.)

lim [f(x)]^n = [lim f(x)]^n, for any value of n that is a rational number.
x->a x->a

Due to constraints regarding Steam's implementation of textboxes (almost seems like they weren't made for mathematical calculations,) our example will unfortunately be very simple. However, I urge those of you intrigued enough to read to here to attempt problems involving these properties yourself.

Evaluate lim (3x^2+ 4x - 1)
x->2
Solution :
= lim (3x^2) + lim (4x) - lim (1)
x->2 x->2 x->2

= 3 lim (x^2) + 4 lim (x) - 1
x->2 x->2

= 3lim x ^2 +4(2) - 1
x->2

= 3(2)^2+8-1
= 19

When we are solving for limits algebraically, we can use the techniques of:
- Direct Substitution
- Factoring
- Rationalizing
- One-sided limits
- Change of variables

For these techniques, you may also use a graph and/or a table in order to check your result.