LIMITS AT INFINITY Section 3.5.

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Presentation transcript:

LIMITS AT INFINITY Section 3.5

When you are done with your homework, you should be able to… Determine (finite) limits at infinity Determine the horizontal asymptotes, if any, of the graph of a function Determine infinite limits at infinity

Pythagoras lived in 550BC. He was the 1st person to teach that nature is governed by mathematics. Another of his achievements was: The mathematics of musical harmony—2 tones harmonize when the ratio of their frequencies form a “simple fraction”. He taught that atoms were in the shape of regular polyhedra. He started a new religion (secret society) which greatly influenced western religion. A and C.

DEFINITION OF LIMITS AT INFINITY Let L be a real number. The statement means that for each there exists an such that whenever The statement means that for each there exists an such that whenever

THEOREM: LIMITS AT INFINITY If r is a positive rational number and c is any real number, then If is defined when then

Evaluate the following limit.

HORIZONTAL ASYMPTOTES The line is a horizontal asymptote of the graph of f if

GUIDELINES FOR FINDING LIMITS AT OF RATIONAL FUNCTIONS If the degree of the numerator is less than the degree of the denominator, then the limit of the rational function is 0. If the degree of the numerator is equal to the degree of the denominator, then the limit of the rational function is the ratio of the leading coefficients. If the degree of the numerator is greater than the degree of the denominator, then the finite limit of the rational function does not exist.

Find the horizontal asymptote(s) of the following function. 3, -4 2 y = 2 B and C None of the above.

Find the horizontal asymptote(s) of the following function. No horizontal asymptotes.

DEFINITION OF INFINITE LIMITS AT INFINITY Let f be a function defined on the interval The statement means that for each positive number M there is a corresponding number such that whenever 2. The statement means that for each negative number M there is a corresponding number such that

Find the horizontal asymptote(s) of the following function. No horizontal asymptotes.