Question

for the following rational functions, find a.) \\(\lim_{x\to\infty} f(x)\\), b.) \\(\lim_{x\to -\infty} f(x)\\), and c.) the equation of the horizontal asymptote, if it exists.\
12.) \\(f(x) = \frac{x^2 - 4x + 3}{x - 1}\\)\
13.) \\(f(x) = \frac{2x^3 + 10x^2 + 12x}{x^3 + 2x^2}\\)\
14.) \\(f(x) = \frac{x^2 - 9}{x^2(x - 3)}\\)

Explanation:

For 12.) $f(x) = \frac{x^2-4x+3}{x-1}$

Step1: Simplify the rational function

Factor numerator: $x^2-4x+3=(x-1)(x-3)$, so
$f(x)=\frac{(x-1)(x-3)}{x-1}=x-3$ (for $x
eq1$)

Step2: Find $\lim_{x\to+\infty}f(x)$

Substitute simplified function:
$\lim_{x\to+\infty}(x-3)=+\infty$

Step3: Find $\lim_{x\to-\infty}f(x)$

Substitute simplified function:
$\lim_{x\to-\infty}(x-3)=-\infty$

Step4: Check for horizontal asymptote

Horizontal asymptotes exist only if the limit at $\pm\infty$ is a finite number. Since both limits are infinite, no horizontal asymptote exists.

Step1: Simplify the rational function

Factor numerator and denominator:
Numerator: $2x(x^2+5x+6)=2x(x+2)(x+3)$
Denominator: $x^3(x+2)$
Cancel common factors ($x
eq0,-2$):
$f(x)=\frac{2(x+3)}{x^2}$

Step2: Find $\lim_{x\to+\infty}f(x)$

Divide numerator/denominator by $x^2$:
$\lim_{x\to+\infty}\frac{2(\frac{1}{x}+\frac{3}{x^2})}{1}=0$

Step3: Find $\lim_{x\to-\infty}f(x)$

Divide numerator/denominator by $x^2$:
$\lim_{x\to-\infty}\frac{2(\frac{1}{x}+\frac{3}{x^2})}{1}=0$

Step4: Identify horizontal asymptote

Since both limits equal 0, the horizontal asymptote is $y=0$.

Step1: Simplify the rational function

Factor numerator: $x^2-9=(x-3)(x+3)$, so
$f(x)=\frac{(x-3)(x+3)}{x^2(x-3)}=\frac{x+3}{x^2}$ (for $x
eq3$)

Step2: Find $\lim_{x\to+\infty}f(x)$

Divide numerator/denominator by $x^2$:
$\lim_{x\to+\infty}\frac{\frac{1}{x}+\frac{3}{x^2}}{1}=0$

Step3: Find $\lim_{x\to-\infty}f(x)$

Divide numerator/denominator by $x^2$:
$\lim_{x\to-\infty}\frac{\frac{1}{x}+\frac{3}{x^2}}{1}=0$

Step4: Identify horizontal asymptote

Since both limits equal 0, the horizontal asymptote is $y=0$.

Answer:

a.) $\boldsymbol{+\infty}$
b.) $\boldsymbol{-\infty}$
c.) No horizontal asymptote exists.

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For 13.) $f(x) = \frac{2x^3+10x^2+12x}{x^4+2x^3}$