MATH 1170 Section 2.3 Worksheet
NAME
Important Note: The ideas that we will be discussing on this worksheet are very complex. It will take reading through sentences multiple times in order for their meaning to register. I suggest you do this. The proofs have breaks after each sentence for you to absorb their content (as well as for you to make notes that help you better understand).
Recall our more precise definition of the limit of a function:
Let f be a function defined on some open interval that contains the number a, except possibly at a itself at a itself. Then we say that the limit of f(x) as x approaches a is L, and we write
lim f(x) = L x→a if every number > 0 there is a corresponding number δ > 0 such that if 0 < |x − a| < δ then |f(x) − L| < .
We have already seen that estimating limits using graphs and tables can be problematic. Hence, it would help if we were able to derive some tools to help us compute limits when they appear to be more complicated.
Limit Laws
As responsible investigators, we will attempt to establish each of these limit laws. But, don’t worry, we are going to walk through the proofs of a few of the Laws of Limits together.
Sum Law
The first Law of Limits is the Sum Law. The Sum Law basically states that the limit of the sum of two functions is the sum of the limits.
The Sum Law
If limx→a f(x) = L and limx→a g(x) = M both exist then
lim[f(x) + g(x)] = L + M. x→a
However, before we can walk through the proof of this law, let’s establish what is called the Triangle Inequality.
1 The Triangle Inequality
If a and b are any real numbers, then
|a + b| ≤ |a| + |b|.
proof of the Triangle Inequality: Assume that a and b are real numbers. What can be said about the relationship between |a + b| and |a| + |b| if a and b are both positive?
Similarly, what can be said about the relationship between |a + b| and |a| + |b| if both a and b are both negative?
Notice that since either a = |a| or a = −|a|, −|a| ≤ a ≤ |a|. Similarly, since either b = |b| or b = −|b|, −|b| ≤ b ≤ |b|. What do you get when you add these inequalities?
Apply the fact that |x| ≤ c if and only if −c ≤ x ≤ c to your result.
What have we established?
Now we are ready to start discussing the proof of the Sum Law. But before we dive right into the proof, we want to make some preliminary notes.
2 Preliminary Notes: In order to prove this law, what we want to establish:
given that lim f(x) = L and lim g(x) = M both exist, x→a x→a we want to be able to show lim[f(x) + g(x)] = L + M. x→a
According to our precise definition of a limit, getting lim[f(x) + g(x)] = L + M means that for any > 0, x→a
we must find a δ > 0 such that if 0 < |x − a| < δ then < .
Using the Triangle Inequality, we can write (*)