Calc 1 Lecture Notes
Section 1.6
Page 1 of 7
Section 1.6: Formal Definition of a Limit
Big Idea: In this section, we formalize the idea that a limit is the number to which a sequence of everbetter approximations converge.
Big Skill: You should be able to prove a limit that is a real number or that involves infinity.
It should be no surprise by now that lim  x 2  1  1 . You have seen how to verify this numerically by
x 0
computing values of x2 + 1 for x-values close to 0. In this section, we are essentially going to invert
our calculations by picking y-values close to 1, and then trying to see if we can find x-values that
produce that y-values. If we can always find an x-value no matter how close our y-values are to the
limit, then we formally will have proved the limit.
x
0.10
0.05
0.02
0.01
0.001
f(x)
1.01
1.0025
1.0004
1.0001
1.000001
 = f(x) – 1
0.01
0.0025
0.0004
0.0001
0.000001
This table numerically supports the fact that
lim  x 2  1  0 ; the closer x gets to 0, the smaller
x 0
This picture graphically supports the fact that
lim  x 2  1  1 .
the difference between f(x) and the limit.
x 0
Repeat of the big idea: if we want to prove that a certain number L is the limit of a function f(x) as
x approaches a, then we must prove that for any y value arbitrarily close to L, there are x values
arbitrarily close to a that generate those y values.
So, looking at the table above,
 y values that are within 0.01 of the limit of the function require |x|  0.1.
 y values that are within 0.0004 of the limit of the function require |x|  0.02.
 y values that are within 0.000001 of the limit of the function require |x|  0.001.
Instead of just making a big list of examples, we can generalize the pattern above using a variable:
 y values that are within  of the limit of the function require |x|   .
The symbol  (”epsilon”) is frequently used in mathematics to represent an arbitrarily small number.
In this section, it will refer to the small distance between a given y value and the limit.
The symbol  (“delta”) also is used to represent an arbitrarily small number. In this section, it will
refer to the small distance between x values.
Calc 1 Lecture Notes
Section 1.6
Page 2 of 7
Practice:
1. We know that lim 2 x  1  5 . Find the values for x for which (2x – 1) is within distance
x 3
1
of 5.
10
1
. State the functional relationship that delta (distance from 3 in the
100
x direction) has on epsilon (distance from 5 in the y direction).
Repeat for a distance of
Calc 1 Lecture Notes
Section 1.6
Page 3 of 7
2. lim x2  1  5 . Find the values for x for which (x2 + 1) is within distance
x 2
1
of 5. Repeat for a
10
1
. State the functional relationship that delta (distance from 2 in the x direction)
100
has on epsilon (distance from 5 in the y direction).
distance of

y















x
Calc 1 Lecture Notes
Section 1.6
Page 4 of 7
Definition 6.1: Precise definition of a Limit
For a function f defined in some open interval a (but not necessarily at a itself), we say lim f  x   L ,
x a
if given any (tiny) number  > 0, there is another number  > 0 such that 0 < |x – a| <  guarantees that
|f(x) - L| <  .
To formally prove a limit lim f  x   L using the book’s technique:
x a

start with f  x   L   ; substitute the actual function definition in for f.


“massage” the inequality until it reduces to |x – a| < some expression involving .
the expression involving  is your .
Alternative technique:

Solve the following two equations for x: f(x) = L +  and f(x) = L –  .

Compute  for both solutions by taking  = |x – a|.

Choose the smaller of the two answers for .
Practice:
3. Use the precise definition of limit to prove lim 2 x  1  5 .
x 3
y





   






x


4. Use the precise definition of limit to prove lim x2  1  5 .
x 2

y















x
Calc 1 Lecture Notes
Section 1.6
Page 5 of 7
5. Use the precise definition of limit to prove lim  x3  1  1.
x 0
Definition 6.2 (Precise Definition of a Limit that Tends to Infinity):
For a function f defined in some open interval containing a (but not necessarily at a itself), we say
lim f  x    if given any number M > 0, there is another number  > 0 such that 0 < |x – a| < 
x a
guarantees that f(x) > M.
To formally prove a limit lim f  x    using the book’s technique:
x a

Start with f(x) > M; substitute the actual function definition in for f.

Manipulate the inequality until it reduces to |x – a| < (an expression involving M).

the expression involving M is your .
Alternative technique:

Solve the following equation for x: f(x) = M.

Compute by taking  = |x – a|.

If there are two values for , choose the smaller value.
Practice:
6. Prove that lim
x 0
1
.
x2
Calc 1 Lecture Notes
Section 1.6
Page 6 of 7
Definition 6.3 (Precise Definition of a Limit that Tends to Negative Infinity):
For a function f defined in some open interval containing a (but not necessarily at a itself), we say
lim f  x    if given any number N < 0, there is another number  > 0 such that 0 < |x – a| < 
x a
guarantees that f(x) < N.
To formally prove a limit lim f  x    using the book’s technique:
x a

Start with f(x) < N; substitute the actual function definition in for f.

Manipulate the inequality until it reduces to |x – a| < (an expression involving N).

the expression involving N is your .
Alternative technique:

Solve the following equation for x: f(x) = N.

Compute by taking  = |x – a|.

If you get two values for, choose the smaller value.
Practice:
7. Prove that lim
x 1
2
  .
x 1
Definition 6.4 (Precise Definition of a Limit at Infinity):
For a function f defined on an interval (a, ) for some a > 0, we say lim f  x   L if given any number
x 
 > 0, there is another number M > 0 such that x > M guarantees that f  x   L   .
To formally prove a limit lim f  x   L using the book’s technique:
x 

Start with f  x   L   ; substitute the actual function definition in for f.

Manipulate the inequality until it reduces to x > M.
Alternative technique:

Solve one of the following two equations for M: f(M) = L +  or f(M) = L –  (one will be
solvable, and one will (probably) not be solvable).

If you get two values for M, choose the larger value.
Calc 1 Lecture Notes
Section 1.6
Page 7 of 7
Practice:
8. Prove that lim
x 
1
 0.
x2
Definition 6.5: Precise Definition of a Limit at negative infinity
For a function f defined on an interval (-, a) for some a < 0, we say lim f  x   L if given any
x 
number  > 0, there is another number N < 0 such that x < N guarantees that f  x   L   .
To formally prove a limit lim f  x   L using the book’s technique:
x 

Start with f  x   L   ; substitute the actual function definition in for f.

Manipulate the inequality until it reduces to x > M.
Alternative technique:

Solve one of the following two equations for N: f(N) = L +  or f(N) = L –  (one will be
solvable, and one will (probably) not be solvable).

If you get two values for N, choose the smaller value.
Practice:
x 1
1.
x  x  1
9. Prove that lim