Limits and Continuity Definition Evaluation of Limits Continuity

Limits, Graphs, and Calculators Graph 1 Graph 2

c) Find 6 Note: f (-2) = 1 is not involved -2

3) Use your calculator to evaluate the limits Answer : 16 Answer : no

One-Sided Limits The right-hand limit of f (x), as x approaches a, equals L

The left-hand limit of f (x), as x approaches a, equals M written: if

Examples of One-Sided Limit 1. Given Find

A Theorem This theorem is used to show a limit does not exist. For

Indeterminate Forms Indeterminate forms occur when substitution in the limit results in 0/0. In

Continuity A function f is continuous at the point x = a if the

A function f is continuous at the point x = a if the following

Examples At which value(s) of x is the given function discontinuous? Continuous everywhere except

and Thus h is not cont. at x=1. h is continuous everywhere else an

Continuous Functions If f and g are continuous at x = a, then A

Intermediate Value Theorem If f is a continuous function on a closed interval [a,

Example f (x) is continuous (polynomial) and since f (1) < 0 and f

Limits at Infinity For all n > 0, provided that Ex. is defined. Divide

Infinite Limits For all n > 0, More Graphs

Limit and Trig Functions From the graph of trigs functions we conclude that they

Tangent and Secant Tangent and secant are continuous everywhere in their domain, which is

Limit and Exponential Functions The above graph confirm that exponential functions are continuous everywhere.

Examples Find the asymptotes of the graphs of the functions

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Limits and Continuity • Definition • Evaluation of Limits • Continuity • Limits Involving Infinity

Limit L a

Limits, Graphs, and Calculators Graph 1 Graph 2

Graph 3

c) Find 6 Note: f (-2) = 1 is not involved -2

3) Use your calculator to evaluate the limits Answer : 16 Answer : no limit Answer : 1/2

The Definition of Limit L a

Examples What do we do with the x?

1/2 1 3/2

One-Sided Limits The right-hand limit of f (x), as x approaches a, equals L written: if we can make the value f (x) arbitrarily close to L by taking x to be sufficiently close to the right of a. L a

The left-hand limit of f (x), as x approaches a, equals M written: if we can make the value f (x) arbitrarily close to L by taking x to be sufficiently close to the left of a. M a

Examples of One-Sided Limit 1. Given Find

More Examples Find the limits:

A Theorem This theorem is used to show a limit does not exist. For the function But

Limit Theorems

Examples Using Limit Rule Ex.

More Examples

Indeterminate Forms Indeterminate forms occur when substitution in the limit results in 0/0. In such cases either factor or rationalize the expressions. Ex. Notice form Factor and cancel common factors

More Examples

The Squeezing Theorem See Graph

Evaluate:

Continuity A function f is continuous at the point x = a if the following are true: f(a) a

A function f is continuous at the point x = a if the following are true: f(a) a

Examples At which value(s) of x is the given function discontinuous? Continuous everywhere except at

and Thus h is not cont. at x=1. h is continuous everywhere else an d Thus F is not cont. at F is continuous everywhere else

Continuous Functions If f and g are continuous at x = a, then A polynomial function y = P(x) is continuous at every point x. A rational function at every point x in its domain. is continuous

Intermediate Value Theorem If f is a continuous function on a closed interval [a, b] and L is any number between f (a) and f (b), then there is at least one number c in [a, b] such that f(c) = L. f (b) f (c) = L f (a) a c b

Example f (x) is continuous (polynomial) and since f (1) < 0 and f (2) > 0, by the Intermediate Value Theorem there exists a c on [1, 2] such that f (c) = 0.

Limits at Infinity For all n > 0, provided that Ex. is defined. Divide by

More Examples