Math 2250 Written HW #2 Solutions
1. Evaluate the following limit or explain why it doesn’t exist:
√
x−3
lim
.
x→9 x − 9
Answer: Notice that we can rationalize the numerator:
√
√
√
x−3
x−3
x+3
x−9
√
lim
= lim
·√
= lim
.
x→9 x − 9
x→9 x − 9
x + 3 x→9 (x − 9) ( x + 3)
Canceling the factors of x − 9, the above limit is equal to
lim √
x→9
since
√
1
1
=
6
x+3
x + 3 goes to 6 as x goes to 9.
2. Draw the graph of a function g(x) for which each of the following is true:
· lim g(x) = 4
x→−2
· g(−2) = 1
· lim g(x) = g(0) = 2
x→0
· lim g(x) = g(1) = 0
x→1−
· lim g(x) = 3
x→1+
Answer: Of course, there are many functions satisfying these conditions. Here is the graph
of one:
4
3
2
1
-3
-2
-1
1
-1
-2
1
2
3
3. Einstein’s theory of special relativity says that the apparent length of a spaceship depends on
how fast the spaceship is moving relative to an observer (this is known as “length dilation”).
Specifically, if the observer measures the spaceship’s length to be L0 when the spaceship is at
rest, then the length of the spaceship (as a function of velocity) is
r
v2
L(v) = L0 1 − 2 ,
c
where c is the speed of light. What happens to the apparent length of the spaceship as its
velocity approaches the speed of light from below? (Hint: write this as a one-sided limit,
then evaluate the limit.)
Answer: The hard part of this question is just to translate what it’s asking into mathematical
language. Since we’re interested in what happens as the speed approaches the speed of light
from below, we surely want to evaluate some limit of the form
lim F (v)
v→c−
for some function F of the variable v. Of course, the function we’re interested in is the length
function, so in fact we want to evaluate
r
v2
lim L(v) = lim L0 1 − 2 .
c
v→c−
v→c−
Now,
as v → c− , we know that v 2 goes to c2 , so v 2 /c2 gets very close to 1, and hence
p
1 − v 2 /c2 is approaching zero (notice that this is why it’s important that we’re taking the
one-sided limit; if v were approaching c from above, v 2 /c2 would be bigger than 1, and we’d
be trying to take the square root of a negative number). Therefore,
r
v2
lim L0 1 − 2 = 0.
c
v→c−
In other words, as the velocity of the spaceship approaches the speed of light, the spaceship
will appear to be getting very short and squat to some outside (stationary) observer. Of
course the people inside the spaceship will perceive the spaceship as being the same size it
always was.
2