We now state some properties of limits without prools.

Assume that lim f (x) and lim g (x) exist. Let c be any real number.
                    x→a           x→a
Then the following are true:

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

   That is, the limit of sum or difference is the sum or difference of the limits.

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

That is, the limit of a product is the product of limits
                      lim f (x)
3.  lim f (x) = x  a       , provided lim g (x) ≠  0.
     x-a  g (x)   lim g (x)                x→ a
                    x →a

     That is, the limit of a quotient is the quotient of the limits, provided the denominator
     does not have a limit of 0.

4.    lim [cf (x)] = c. lim f (x)
     x→ a                     x → a

That is, the limit of constant times a function is the constant times the limit.

5.    If h (x) = c for all x, then lim h (x) = c.
                                               x →a

That is, the limit of a constant function is a constant. This is usually abbreviated by writing lim c = c.
                                                                                                                                  x→ a
  6. lim   for any positive integer n.

  7. lim xⁿ = aⁿ for any positive integer n.

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