The Tiger is Hungry!
You look up from your Mathematics lesson to be confronted by a hungry tiger! You are immediately
frightened (more of the tiger than your maths), so you listen carefully when he talks to you and demands the
square root of 99.
As your feelings of amazement and panic subside, you suddenly remember: YOUR iPHONE HAS A
BUILT-IN CALCULATOR! So you punch up 99 and hit the SQRT (square root) button. The display shows
9
.
9
4
9
8
7
As quick as you can you toss 9.9487 to the Tiger.
The tiger swallows the 9.9487 but immediately demands (with hunger still showing in his eyes) the
square root of 100.3, You silently thank your Aunt Emmy for giving you the iPhone as you ask it to
compute sqrt(100.3). The display reads
1
0
.
0
1
5
0
The Tiger eats 10.015 immediately but now wants the square root of 99.85.
From the looks of things he wants it now! Your iPhone tells you that sqrt(99.85) is
9
.
9
9
2
5
0
which momentarily satisfies the Tiger. But only momentarily. This Tiger's appetite seems predictable yet
insatiable!
He demands the square roots of more and more numbers, each closer to 100 than the last. Your responses
get closer and closer to 10 until the Tiger asks for sqrt(100.001). At this point your display reads
1
0
.
0
0
0
0
Thereafter, the Tiger keeps demanding the square roots of numbers closer to 100 but is content with a
steady diet of 10's for answers.
Your traumatic experience with the Tiger illustrates the concept of a limit. How?
1.
The Tiger supplied numbers which were increasingly close to a fixed number a, called the objective.
The Tiger's objective was 100.
2.
You played the role of a function. You took numbers from the Tiger and gave back their square roots,
so you were the square root function.
3.
The values which you returned approached a fixed number or LIMIT (or value of the limit). Your
limit had value 10.
The first three features can be qualitatively expressed by saying that "as x got closer to 100, the sqrt(x) got
closer to 10". This is the concept of a limit. The concept of limit treats the behavior of a function near but
not at the objective.
The next feature of the Hungry Tiger was that
4.
The Tiger supplied numbers near his objective a=100 but never asked to be fed the square root of a
itself.
The value of the square root function at the point a was not the issue here!
Finally, the Tiger experience had a fifth noteworthy feature.
When the Tiger asked for sqrt(99), throwing out the number 10 would not have sufficed.
10 may be close to sqrt(99), but it is not close enough for that hungry Tiger.
On the other hand, the Tiger did not demand the exact sqrt(99), which would have been impossible to
deliver since it is irrational.
5.
The Tiger had a notion of acceptable error. Your iPhone had limited accuracy,
so the numbers you fed the Tiger were not exactly the actual square roots of the numbers given.
But the Tiger tolerated this error, and once his numbers were sufficiently close to the objective a,
all of the actual square roots were tolerably close to the L of 10.
Adapted from http://oregonstate.edu/instruct/mth251/cq/Stage3/Lesson/limitConcepta.html