L06 - Example Let f x = √ x 1 Evaluate 1 f(0 = 2 lim x...

Info iconThis preview shows pages 1–12. Sign up to view the full content.

View Full Document Right Arrow Icon
Lecture 6 — The Limit of a Function An Introduction Example: Let f ( x ) = x 2 - 9 x - 3 Consider the following table of values for our func- tion: x 2.9 2.99 2.999 3 3.001 3.01 3.1 f ( x ) 5.9 5.99 5.999 6.001 6.01 6.1 NOTE:
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
To see what is happening graphically:
Background image of page 2
THE LIMIT OF A FUNCTION Definition: In our example, we write Example: For f ( x ) = x 2 - 1 , find lim x 1 f ( x ) .
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
We have the following picture: But consider Examples: f (3) = lim x 3 f ( x ) f ( - 1) = lim x →- 1 f ( x )
Background image of page 4
When the limit at a Point does not exist Example: Let f ( x ) = 1 x - 3 . What is lim x 3 f ( x ) ? We can try to find the answer in two ways: 1) Consider the following table of values: x 2.9 2.99 2.999 3 3.001 3.01 3.1 f ( x ) -10 -100 -1000 1000 100 10 Find lim x 3 f ( x )
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
2) We can also see this from the graph of f ( x ) = 1 x - 3 1) lim x 3 f ( x ) = 2) lim x 3 - f ( x ) = 3) lim x 3 + f ( x ) =
Background image of page 6
Example: Let f ( x ) = ± 2 x < 0 x 2 x 0 Its graph: Find: 1) f (0) 4) lim x 0 + f ( x ) 2) lim x 0 f ( x ) 5) lim x 2 f ( x ) 3) lim x 0 - f ( x )
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Example: Evaluate: 1) lim x →- 3 f ( x ) 5) lim x 2 f ( x ) 2) f (0) 6) lim x 2 - f ( x ) 3) lim x 0 f ( x ) 7) lim x 2 + f ( x ) 4) f (2)
Background image of page 8
TO EVALUATE LIMITS: Algebraic Methods:
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Background image of page 10
Background image of page 11

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Background image of page 12
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Example: Let f ( x ) = √ x + 1 . Evaluate: 1) f (0) = 2) lim x → √ x + 1 NOTE: Some Basic Limits 3) lim x → a b = 4) lim x → a x = 5) lim x → a x n = 6) lim x → a n √ x = for appropriate values of x and a . Properties of Limits Let L := lim x → a f ( x ) and M := lim x → a g ( x ) . Then for any real numbers c and r , 1) lim x → a c · f ( x ) = 2) lim x → a h f ( x ) ± g ( x ) i = 3) lim x → a h f ( x ) · g ( x ) i = 4) lim x → a f ( x ) g ( x ) = 5) lim x → a [ f ( x ) ] r = NOTE: If p ( x ) is a polynomial, then lim x → a p ( x ) = Example: lim x →-1 x 2 + 2 x-3 = Example: lim x →-3 x 2 + 3 x x-3 = Example: lim x → 3 x 2-9 x-3 =...
View Full Document

{[ snackBarMessage ]}

Page1 / 12

L06 - Example Let f x = √ x 1 Evaluate 1 f(0 = 2 lim x...

This preview shows document pages 1 - 12. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online