Applied Calculus: UNIT I: Differential Calculus
Asymptotes
Limit of a function | Differential Calculus
Explanation, Formula, Equation, Example and Solved Problems - Differential Calculus: Limit of a function: Asymptotes
ASYMPTOTES
Unlike polynomials whose graphs are continuous (unbroken) curves, the graphs of rational functions have discontinuities at the points where the denominator is zero. Unlike polynomials, rational functions may have numbers at which they are not defined. Near such points, many rational functions have graphs that closely approximate a vertical line, called a vertical asymptote.
HORIZONTAL ASYMPTOTE
Unlike the graphs of non‒constant polynomials, which eventually rise or fall indefinitely, the graphs of many rational functions eventually get closer and closer to some horizontal line, called a horizontal asymptote.
Example 74. Find the horizontal and vertical asymptotes of the graph of the function
Solution: Dividing both numerator and denominator by x and using the properties of limits, we have
Therefore the line y = √2/3 is a horizontal asymptote of the graph of f.
In computing the limit as x → ‒∞, we must remember that for x < 0, we have √x2 = |x| = ‒x. So when we divide the numerator by x, for x < 0 we get
Thus the line y = −√2/3 is also a horizontal asymptote.
A vertical asymptote is likely to occur when the denominator, 3x‒5, is 0, that is, when x = 5/3. If x is close to 5/3 and x > 5/3, then the denominator is close to 0 and 3x‒5 is positive. The numerator √[2x2+1] is always positive, so f(x) is positive. Therefore
If x is close to 5/3 but x < 5/3, then 3x‒5< 0 and so f(x) is large negative. Thus
The vertical asymptote is x = 5/3. All three asymptotes are shown in Figure 8.
Applied Calculus: UNIT I: Differential Calculus : Tag: Applied Calculus : Limit of a function | Differential Calculus - Asymptotes
Applied Calculus: UNIT I: Differential Calculus
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