section 2.1 introduction to limits. definition of a limit limits allow us to describe how the...
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Section 2.1INTRODUCTION TO LIMITS
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Definition of a Limit
Limits allow us to describe how the outputs of a function (usually the y or f(x) values) behave as the inputs (x values) approach a particular value.
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Limit Notation
The previous definition is confusing, but all it really says is that a function has a limit at a particular value if the function doesn’t go crazy in the vicinity of that value.
In other words, if you only look at the x-values near the value you are trying to find a limit for, can you graph all of the nearby y-values in a window?
If you can, then we use the notation: . This means that the function approaches L as x
approaches c.
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One and Two Sided Limits
When we say that the function “approaches” a particular value, it can do so moving from the left, or from the right.
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Another way to think of limits
A function f(x) has a limit as x approaches c if and only if the right-hand and left-hand limits at c exist and are equal. In other words the function must be approaching the same value from both sides.
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Example
Graph the following function in your calculator.
Compare the limits and the values of the function at various spots on the graph.
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Do-Now
Greatest Integer Function (Int x): The function for which…..Input: all real numbers x.Output: The largest integer less than or
equal to x. Sketch a graph for this function and complete
pg 63 #37-40.
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Finding limits algebraically
Graph the following functions in your calculator.
1. 2.
What do the graphs of these functions tell you about the limit of the function as x approaches 3?
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Limits of Rational Functions
Can you find the limit as x approaches 3 by using direct substitution?Why or why not?
Why did the limit not exist in #1 but it did in function #2?
Use algebra to simplify the expressions and confirm the limits that you found graphically.
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Properties of Limits
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Properties of Limits Continued
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Calculator exercise:
Find: Find: Knowing these limits can allow you
to find other limits algebraically.
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Examples
1. 2. 3. 4. (hint: change tan x to something
else)
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14 2014 – APSI – Day 1
Key Limits that are helpful to know
1lim 0x x
0
1limx x
0
1limx x
0
sinlim 1x
xx
0
1 coslim 0x
xx
1
0lim 1 x
xx e
1lim 1
x
xe
x
0
1lim ( ) limx x
f x fx
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Sandwich Theorem