Limits and Continuity Definition Evaluation of Limits Continuity Limits Involving Infinity

Limit L a

Limits, Graphs, and Calculators c) Use direct substitution **If direct substitution results in an indeterminate form, then try factoring or rationalizing to simplify f(x) and then try direct substitution again, Indeterminate form

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

3) Use direct substitution to evaluate the limits Answer : 16 Answer : DNE Answer : DNE Answer : 1/2

One-Sided Limit 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 Find

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

Limit Theorems Page 99

Examples Using Limit Rule

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

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The Squeezing Theorem See Graph

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) Memorize a

Removable Discontinuities: (You can fill the hole.) Essential Discontinuities: infinite oscillating jump

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

F is continuous everywhere else h is continuous everywhere else and and Thus F is not cont. at Thus h is not cont. at x=1. F is continuous everywhere else h 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 is continuous at every point x in its domain.

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. Memorize 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 is defined. Divide by Ex.

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Infinite Limits For all n > 0, More Graphs

Examples Find the limits

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

Tangent and Secant Tangent and secant are continuous everywhere in their domain, which is the set of all real numbers

Inverse tangent function f(x)=tan x is not one-to-one But the function f(x)=tan x , -π/2 < x < π/2 is one-to-one. The restricted tangent function has an inverse function which is denoted by tan-1 or arctan and is called inverse tangent function. Example: tan-1(1) = π/4 . Limits involving tan-1:

Graphs of inverse functions The graph of y = arctan x Domain: Range:

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

Asymptotes

Examples Find the asymptotes of the graphs of the functions