ThE LIMIt OF A FUNCTION EVALUATING LIMITS – THE LIMIT LAWS Let's quickly reflect on what we covered in the previous section. We tried to evaluate limits using numerical and graphical methods, but we saw that these methods could only be used to make approximations. We also found that there are a number of pitfalls to guessing the values of limits, which usually leads to making wrong guesses. In this section, we will use foolproof methods to compute the EXACT values of limits. We will also examine ➔ how to combine the various laws to evaluate seemingly complicated limits, and ➔ Theorems that ease simplification of limits We start with the limit laws. The textbook I'm using highlights eleven limit laws. The first five deal with basic limit operations: addition, subtraction, multiplication, division and multiplication by a constant. The first five laws are stated below. Note that all eleven laws are valid if and only if the limits
lim f(x)
lim g(x)
and
x→a exist, and also if c is a constant:
x→a
Law 1 – Addition Law:
lim
x→a
[f(x)
+ g(x)]
lim f(x)
+
lim f(x)
–
=
x→a
lim g(x)
x→a
Law 2 – Difference Law:
lim
x→a
[f(x)
– g(x)]
=
x→a
lim g(x)
x→a
Law 3 – Constant Multiple Law:
lim
x→a
[c f(x)]
=
c lim f(x) x→a
Law 4 – Product Law:
lim
x→a
[f(x) g(x)]
lim f(x)
=
x→a
×
lim g(x)
x→a
Law 5 – Quotient/Division Law:
lim
x→a
f(x) g(x)
=
lim f(x)
x→a
lim g(x)
x→a
[On the condition that the denominator IS NOT ZERO]