3.5: ASYMPTOTES.

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Presentation transcript:

3.5: ASYMPTOTES

VERTICAL ASYMPTOTES If f (x) approaches infinity (or negative infinity) as x approaches c from the right or from the left, then the line x = c is a vertical asymptote of the graph of f.

Look at the graph of

EXAMPLES Evaluate the following limits:

MORE THAN ONE ASYMPTOTE Find the vertical asymptote(s) of the graph of

EXAMPLE Find the vertical asymptote(s) of the graph of What happens at x = -3 ?

EXAMPLE Find the limits:

HORIZONTAL ASYMPTOTES If f is a function and L1 and L2 are real numbers, the statements below denote limits at infinity. The lines y = L1 and y = L2 are horizontal asymptotes of the graph of f.

FIND THE LIMIT:

HORIZONTAL ASYMPTOTES Let f (x) = p (x)/q (x) be a rational function. If the degree of the numerator is less than the degree of the denominator, then y = 0 is a horizontal asymptote of the graph of f. If the degree of the numerator is equal to the degree of the denominator, then y = a/b is a horizontal asymptote of the graph of f, where a and b are the leading coefficients of p(x) and q(x), respectively. If the degree of the numerator is greater than the degree of the denominator, then the graph of f has no horizontal asymptote.

FIND THE HORIZONTAL ASYMPTOTE OF THE GRAPH OF THE FUNCTION: