5.2 Limit of a Function5.2 Limit of a Function

Limit of a Function

5.2 Limit of a Function5.2 Limit of a Function

• The concept of a limit is central to the study of calculus.

• Differentiation and integration are each based on the notion of a limit.

• Limit of f (x) as x approaches a – the value that f (x) approaches as x gets closer to a

• In symbols:

€

limx → a

f (x)

5.2 Limit of a Function5.2 Limit of a Function

Observe the values of f (x) = x2 + 1 as x approaches 3:

We say that .

€

limx → 3

f (x) =10

x 2 2.5 2.9 2.99 2.999 3 3.001 3.01 3.1 3.5 4

f (x) 5 7.25 9.41 9.94 9.994 10 10.006 10.06 10.61 13.25

17

5.2 Limit of a Function5.2 Limit of a Function

5.2 Limit of a Function5.2 Limit of a Function

Given . Find the limit of

f (x) as x approaches 3..

We still say that .

€

limx → 3

f (x) =10

x 2 2.5 2.9 2.99 2.999 3 3.001 3.01 3.1 3.5 4

f (x) 5 7.25 9.41 9.94 9.994 15 10.006 10.06 10.61 13.25

17

€

f (x) =x 2 +1 for x ≠ 3

15 for x = 3

⎧ ⎨ ⎩

5.2 Limit of a Function5.2 Limit of a Function

Given . Find the limit of

f (x) as x approaches 2..

€

limx → 2−

f (x) = 5

x 1 1.5 1.9 1.99 1.999 2 2.001 2.01 2.1 2.5 3

f (x) 3 4 4.8 4.98 4.998 4 4.004 4.04 4.41 6.25 9

€

f (x) =2x +1 for x < 2

x 2 for x ≥ 2

⎧ ⎨ ⎩

€

limx → 2+

f (x) = 4

5.2 Limit of a Function5.2 Limit of a Function

• The limit of the function f exists only if the left-hand and right-hand limits both exist and are equal.

• exists if and only if

€

limx → a

f (x)

€

limx → a −

f (x) = limx → a +

f (x) = L

5.2 Limit of a Function5.2 Limit of a Function

Limit Theorems

5.2 Limit of a Function5.2 Limit of a Function

L1. The limit of a constant function is the constant.

Examples.

€

limx → a

k = k

€

limx → 5

− 2 = −2

€

limx → −3

32 = 3

2

5.2 Limit of a Function5.2 Limit of a Function

L2. If f is a polynomial function, then

Example.

€

limx → a

f (x) = f (a)

€

limx → 3

(x 3 + 6x) = 33 + 6(3) = 45

5.2 Limit of a Function5.2 Limit of a Function

L3. The limit of a sum is the sum of limits.

Example.

€

limx → a

f (x) + g(x)[ ] = limx → a

f (x) + limx → a

g(x)

€

limx → 2

x 2 + 2x − 3( ) = limx → 2

x 2 + limx → 2

2x − limx → 2

3

= 4 + 4 − 3

= 5

5.2 Limit of a Function5.2 Limit of a Function

L4. The limit of a constant times a function is the constant times the limit of the function.

Example.

€

limx → a

kg(x) = k limx → a

g(x)

€

limx → 5

12 (x + 3) = 1

2 limx → 5

(x + 3) = 12 (8) = 4

5.2 Limit of a Function5.2 Limit of a Function

L5. The limit of a product is the product of the limits.

Example.

€

limx → a

f (x)⋅ g(x)[ ] = limx → a

f (x)[ ] limx → a

g(x)[ ]

€

limx →1

(x + 3)(x 2 +1) = limx →1

(x + 3)⋅ limx →1

(x 2 +1)

= 4⋅ 2

= 8

5.2 Limit of a Function5.2 Limit of a Function

L6. The limit of a quotient is the quotient of the limits.

Example.

€

limx → a

f (x)

g(x)=

limx → a

f (x)

limx → a

g(x), lim

x → a g(x) ≠ 0

€

limx → 2

x − 3

x +1=

limx → 2

(x − 3)

limx → 2

(x +1)= −

1

3

5.2 Limit of a Function5.2 Limit of a Function

L7. The limit of a function raised to an exponent is the limit of the function raised to that exponent.

Example. €

limx → a

f (x)[ ]n

= limx → a

f (x)[ ]n

€

limx → 2

x 2 −1( )3

= limx → 2

x 2 −1( )[ ]3

= 33 = 27

5.2 Limit of a Function5.2 Limit of a Function

Example. Find .

€

limx → 2

x 2 + 8

x 3 − 4

⎛

⎝ ⎜

⎞

⎠ ⎟

3

€

limx → 2

x 2 + 8

x 3 − 4

⎛

⎝ ⎜

⎞

⎠ ⎟

3

€

= limx → 2

x 2 + 8

x 3 − 4

⎛

⎝ ⎜

⎞

⎠ ⎟

3

€

=limx → 2

x 2 + 8( )

limx → 2

x 3 − 4( )

⎡

⎣

⎢ ⎢

⎤

⎦

⎥ ⎥

3

€

=12

4

⎛

⎝ ⎜

⎞

⎠ ⎟3

€

=27

5.2 Limit of a Function5.2 Limit of a Function

Example. Find .

€

limx → 2

x 2 − 4

x − 2

€

limx → 2

x 2 − 4

x − 2

€

=limx → 2

(x + 2)(x − 2)

x − 2

€

=limx → 2

(x + 2)

€

=4

5.2 Limit of a Function5.2 Limit of a Function

Example. Given Find

..

€

limx → 2

f (x)

€

f (x) =x 3 − 4 for x < 2

2x for x ≥ 2

⎧ ⎨ ⎩

€

limx → 2−

f (x)

€

limx → 2+

f (x)

€

= limx → 2−

(x 3 − 4)

€

=4

€

= limx → 2+

2x

€

=4

€

limx → 2

f (x) = 4

5.2 Limit of a Function5.2 Limit of a Function

1.a Find .

€

limx → 3

(x 2 + 7x − 5)

€

limx → 3

(x 2 + 7x − 5)

€

=32 + 7(3) − 5

€

=25

5.2 Limit of a Function5.2 Limit of a Function

1.d Find .

€

limx → −2

(x 4 − 3x 3 + 4 x − 5)

€

limx → −2

(x 4 − 3x 3 + 4 x − 5)

€

=(−2)4 − 3(−2)3 + 4(−2) − 5

€

=27

5.2 Limit of a Function5.2 Limit of a Function

2.c Find .

€

limx → 3 / 2

4x 2 − 9

2x + 3

€

limx → 3 / 2

4x 2 − 9

2x + 3

€

=4 3

2( )2

− 9

2 32( ) + 3

€

=0

5.2 Limit of a Function5.2 Limit of a Function

3.a Find .

€

limx → 4

x 2 −16

2x − 8

€

limx → 4

x 2 −16

2x − 8

€

=limx → 4

(x + 4)(x − 4)

2(x − 4)

€

=limx → 4

x + 4

2

€

=4

5.2 Limit of a Function5.2 Limit of a Function

3.c Find .

€

limx → 2

x 2 − 4

x 2 − x − 2

€

limx → 2

x 2 − 4

x 2 − x − 2

€

=limx → 2

(x + 2)(x − 2)

(x − 2)(x +1)

€

=limx → 2

x + 2

x +1

€

=4

3

5.2 Limit of a Function5.2 Limit of a Function

4.b Find .

€

limx → 2

x 3 + 2x + 3

x 2 + 5

€

limx → 2

x 3 + 2x + 3

x 2 + 5

€

= limx → 2

x 3 + 2x + 3

x 2 + 5

€

=15

9

€

=15

9

€

=13 15

5.2 Limit of a Function5.2 Limit of a Function

5.a Suppose and .

Find ..€

limx → 4

f (x) = 0

€

limx → 4

g(x) = −2

€

limx → 4

(g(x) + 2)

€

limx → 4

(g(x) + 2)

€

=limx → 4

g(x) + limx → 4

2

€

=−2 + 2

€

=0

5.2 Limit of a Function5.2 Limit of a Function

7. Given . Find

..

€

limx → 2

f (x)

€

f (x) =x 5 −12 for x < 2

(x + 3)3 − 8 for x ≥ 2

⎧ ⎨ ⎩

€

limx → 2−

f (x)

€

limx → 2+

f (x)

€

= limx → 2−

(x 5 −12)

€

=20

€

= limx → 2+

(x + 3)3 − 8[ ]

€

=19

€

limx → 2

f (x) does not exist.