Section 2.4 - The Precise Definition of a Limit

Consider the function

If x is close to 2 but not equal 2, then f(x) is close to 1. The concept of a limit is that f(x) can be made "arbitrarily" close to 1, by choosing x "close enough" to 2.

Consider being within 0.1 of 1, how close should we get to 2?

Note: Asking that requires that |x - 2| < ??

Equivalent to .9 < f(x) < 1.1 requires that 2 - ? < x < 2 + ?

What if wished to be within 0.01 of 1, how close should x be to 2?

…. Or if wished to be within 0.001, ???

Generalize - If wish be within of f(x), what is that value that we must be within on x.

Definition

Let f be a function defined on an some open interval that contains the number a, except possibly at a itself. Then we say that the limit of f(x) as x approaches a is L, and we write if for every number > 0 there is a corresponding number > 0 such that

whenever

Equivalently:

whenever a - < x < a +

Graphical Interpretation of tolerance

x Î (a - , a + ) implies y Î (L - , L + )

Example:

#2 (transparency)

Proofs can be very "messy", "creative", "tricky", but if f(x) is linear, proofs are "pretty straight forward"

Doing a proof of a linear limit

/.. depends on , so "work backwards" .. then reverse

Text calls this “preliminary analysis (guessing a value for ), then the formal proof

Example: Prove using the Definition of Limit

Means: for every > 0, there is a > 0 such that if 0 < |x - (-3)| < , then |(2x + 1) - (-5)| <

Work backwards to determine appropriate :

|(2x + 1) - (-5)| <

Implies |2x + 6| <

Or |2(x + 3)| <

So 2|x +3| <

Or |x - (-3)| < /2 Thus could let = /2

NOW THE FORMAL PROOF

Given > 0, choose = /2. If 0 < |x - (-3)| < , then

|(2x + 1) - (-5)| = |2x +6| = 2|x + 3| < 2 = 2(/2) =

|(2x + 1) - (-5)| <

Read Definition of Left-Hand Limit, and Right-Hand Limit on page 91

Graphical Example of a "non-linear"

Consider at x = 1

Visual interpretation of "within epsilon of 2", and "within delta of 1"

Assignment: 2.4 – pg 95 – 1, 3, 15-21