Math 400 (Section 2.7)
Exploring Precise Definitions of Limits
Name ________________________________
Recall that we have informally defined limit as follows:
“L is the limit of f(x) as x approaches c if and only if L is the one number that f(x)
can be kept arbitrarily close to just by keeping x close enough (but not equal) to c.”
In this exploration, you will discover and use a more precise definition of limit.
1. What does x mean? _________________________________________________________
2. What does x − c mean? ______________________________________________________
3. Express the inequality A < 3 without the absolute value symbol.
____________________________
4. Write an expression to convey that “the distance between f ( x ) and L is less than ε .”
___________________________
5. Write an expression to convey that “x is within δ units of c, but not equal to c.”
___________________________
Precise Definition of Limit
L = lim f ( x ) if and only if, for any positive ε (no matter how small), there exists some
x→c
positive δ such that f ( x ) − L < ε whenever 0 < x − c < δ .
Discuss with your group members why this definition is exactly equivalent to the verbal
definition given at the top of this page. Make any notes to yourself here.
Mathematicians like to write things as concisely (and as precisely) as possible, especially
when writing notes to themselves. For example, the definition just given can be written like
this:
L = lim f ( x ) iff ∀ε > 0, ∃δ > 0 such that 0 < x − c < δ ⇒ f ( x ) − L < ε .
x→c
6. Graph the function f ( x ) =
lim f ( x ) ?
x →3
2
x + 2 at right. What is
3
______
7. How close must x be kept to 3 in order for f ( x ) to be
within 0.1 units of the limit found in problem 6?
______
 12 x + 2, x < 2
8. Graph the function f ( x ) =  3
x>2
 2 x,
Let L = lim f ( x ) . What is L ? ________
x→2
9. Let ε = 1. We wish to find δ > 0 such that f ( x ) is
within ε of the limit L whenever x − 2 < δ . To help
with this, graph the lines y = L + ε and y = L − ε .
10. What would be a good choice for δ ? Why not choose 2 for δ ?
11. To formally prove that the value of L from problem 8 is in fact lim f ( x ) , we follow these
x→2
steps: (i) Let ε > 0 . (That is, let ε be an arbitrarily small positive number). (ii) Use the
behavior of the function f in the immediate vicinity of 2 to find a good candidate for δ .
(Note that this δ will be expressed in terms of ε .) (iii) Show that the δ you found in step (ii)
satisfies definition of limit. That is, show that
f ( x ) − L < ε whenever 0 < x − 2 < δ .
Prove that lim f ( x ) = ______.
x→2
12. We saw in Problem 10 that if we don't have symmetry about the line x = c , then we should
use the graph on the _________ side of the limit point to find a good candidate for δ . Use
this idea for the function g ( x ) = ( x − 2 ) + 1 to help prove that lim g ( x ) = 5 .
2
x→4