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Some Limit Rules Before Beginning Trig Limits:
- The limit of a constant is the constant
- The limit of sum/difference is the sum/difference of the limits.
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- The limit of product is the product of the limits.
- The limit of a quotient is the quotient of the limits, provided the
limit of the denominator is not 0.
- The limit of a constant times a function is the constant times the limit of
the function.
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Limits of Trigonometric Functions
Ex 1 : Evaluate
a. b.
Ex 2 : Are there any values of c for which or
does not exist? Explain.
Before we begin, please make sure your calculator is in radian mode.
The Squeeze Theorem
If two functions squeeze together at a particular point, then any function
trapped between them will get squeezed to that same point.
The Squeeze Theorem deals with limit values, rather than function values.
In the graph below, the upper and lower functions have the same limit value
at x = a. The middle function has the same limit value because it is trapped
between the two outer functions.
Suppose f(x) < g(x) < h(x) for all x in an
open interval about a (except possibly at a
itself). Further suppose
Then,
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Ex 5 :
Ex 6 :
Ex 7 :
Compute all the following limits without a calculator. (handout)
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Ex 8 :
Ex 9 :
Ex 10 :
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Ex 11 :
Ex 12 :
Ex 13 :
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Ex 14 :
Ex 15 :
Ex 16 :
Homework :
Limits of Trig
Functions Handout
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Ex 1 : Find the derivative of y = 2 x + 3 using the limit definition.
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Ex 2 : Find f '( x ) for f ( x ) = x
2
- 2 x using the limit definition.
Ex 3 : Write the equation of the line tangent to f(x) = x
2
at x = 4.
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Ex 4 : Find dy/dx for y = 1/x.
Ex 5 : Write the equation of the tangent line to f(x) = 1/x at x = 2.
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The definition of the derivative that we have used thus far produces
a derivative function that has to then be evaluated at a specific
value of x. If we know this specific value in advance, then we can
evaluate a less complicated limit that will produce a numeric value
for the derivative immediately.
The Derivative at a Point
To find the derivative of f ( x ) at x = c ,
evaluate:
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Ex 3 : Find the derivative of f ( x ) = √ x + 1 when x = 3.
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Limit Definition of
Continuity
A function is said to be
CONTINUOUS
at x = c if
the twosided limit at x = c exists
the function value at x = c is defined
and these two values are equivalent.
Graphically, when is a function NOT continuous?
Oct 51:45 PM
We cansee if a function is continuous from the graph, but how
do we determine if a function is continuous from the equation?
Ex 1 : Is g(x) continuous when x = 1?
Ex 2 : Is g(x) continuous when x = 1?
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Ex 3 : For what value of k will g(x) be continuous at x = 2?
Homework :
p64 Q1 Q
p74 T2 T
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Differentiability
In order for a function to be differentiable at a specific value of x, the
function must be continuous there and the limit which defines the
derivative must exist.
Geometrically, a function f is differentiable at x
0
if the graph of f has
a tangent line at x
0
. Thus f is not differentiable at any point x
0
where
the secant lines do not approach a unique nonvertical limiting
position as x approaches x
0
. These cases can be described informally
as corner points (cusps) and points of vertical tangency.
What is a cusp?
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Oct 167:06 AM
Determine if the following function is continuous at x = 2:
Is f(x) differentiable at x = 2? Explain.
