Transcript 2.1 Rates of Change and Limits

1.2 Finding limits Graphically and
Numerically
• Estimate a limit using a numerical or graphical
approach.
• Learn different ways that a limit can fail to
exist.
• Study and use a formal definition of limit.
Limit
Gives us a language for describing how the
outputs(y) of a function behave as the inputs(x)
approach some particular value(c).
Informal Definition of a Limit
If f(x) becomes arbitrarily close to a single number L
as x approaches c from either side, then the limit of
f(x) as x approaches c is L, written as
Methods for finding limits at a point c.
1) Graphically- “Meeting at the dinner”
2) Numerically- make a table of values very close to c.
3) Algebraically
• Direct substitution
• Factoring
• Conjugates
• Trigonometric Identities
Graphical- “Meet at the diner”
http://www.calculus-help.com/funstuff/phobe.html
Numerically
Using a table to find the limit.
• Choose the change in the table to be small,
how small?
• Choose to start the table at your c.
Do limits always exists?
A limit at a given value DNE when:
1) The one sided limits approach different values
2) The function increases/decreases without bound
3) The function oscillates between two fixed values
Formal Definition of a Limit
Let f be a function defined on an open interval containing c
(except possibly at c ) and let L be a real number. The
statement
Means that for each ε > 0 there exists a 𝛅 > 0 such that if
0<⃒x-c ⃒ < 𝛅 then ⃒f(x) –L⃒ <𝛆.
What does this mean:
F(x) lies in the interval (L-𝛆, L+𝛆) which is written as
While x lies in either interval (c-𝛅, c) OR (c, c+𝛅) which is written as
To apply the definition you work backwards. Find an 𝛆 that
works and then let 𝛅 =𝛆 so you can prove a limit exists.
Examples