EX #8: Write an equation of a rational function with the following limit properties. A. f(2) is undefined...

Question

EX #8: Write an equation of a rational function with the following limit properties.
A.
f(2) is undefined
lim x→2− f(x) = 3 lim x→2+ f(x) = 3
lim x→−4− f(x) = −∞ lim x→−4+ f(x) = +∞
B.
f(−1) is undefined
lim x→−1− g(x) = 2 lim x→−1+ g(x) = 2
lim x→−3− g(x) = +∞ lim x→−3+ g(x) = +∞

SnapAndSolve AI · Accepted Answer

A. To satisfy the given conditions for f(x):

f(2) is undefined, indicating a removable discontinuity at x = 2. The limits from both sides are equal, suggesting a hole at x = 2. A possible factor is (x-2) in both the numerator and denominator.
The behavior at x = -4 suggests a vertical asymptote, with the function going to -∞ from the left and +∞ from the right. This indicates a factor of (x+4) in the denominator with an odd power.

A possible rational function is:

f(x) = (x-2)(x+4) / (x-2)(x+4)^2

B. To satisfy the given conditions for g(x):

f(-1) is undefined, indicating a removable discontinuity at x = -1. The limits from both sides are equal, suggesting a hole at x = -1. A possible factor is (x+1) in both the numerator and denominator.
The behavior at x = -3 suggests a vertical asymptote, with the function going to +∞ from both sides. This indicates a factor of (x+3) in the denominator with an even power.

A possible rational function is:

g(x) = (x+1)(x+3)^2 / (x+1)(x+3)^2

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EX #8: Write an equation of a rational function with the following limit properties. A. f(2) is undefined...

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