Find Infinite limit (Rational Function)underset (x->5^+)(lim)(1/x^(4/3)-1/(x-5)^(4/3))

True or False. The graph of a rational function may intersect a horizontal asymptote.

Consider the following sets of rational functions. ${\displaystyle af\left(x\right)=\frac{a}{x}}$ for a={−2,−1,−0.5,0.5,2,4}, ${\displaystyle h(x-c)=\frac{1}{x-c}}$​ for c=[−4,−2,−0.5,0.5,2,4], h(x−c)=1x−c for c=[−4,−2,−0.5,0.5,2,4], g(bx)=1bx for b={−2,−1,−0.5,0.5,2,4} for b={−2,−1,−0.5,0.5,2,4}, ${\displaystyle k\left(x\right)+d=\frac{1}{x}+d}$ for d={−4,−2,−0.5,0.5,2,4} for d={−4,−2,−0.5,0.5,2,4}.
c. Choose two functions from any set. Find the slope between consecutive points on the graphs.

Use substitution to convert the integrals to integrals of rational functions. Then use partial fractions to evaluate the integrals. ${\displaystyle \int \frac{\cos x}{\sin x(1-\sin x)}dx}$

Determine ${\displaystyle limx\to \infty f\left(x\right)}$ and ${\displaystyle limx\to -\infty f\left(x\right)}$ for the following rational functions. Then give the horizontal asymptote of f (if any). ${\displaystyle f\left(x\right)=\frac{40x^5+x^2}{16x^4-2x}}$

Involve rational functions that model the given situations. In each case, find the horizontal asymptote as ${\displaystyle x\to \infty }$ and then describe what this means in practical terms.
${\displaystyle f\left(x\right)=\frac{150x+120}{0.05x+1}}$ the number of bass. f(x), months in a lake that was stocked with 120 bass.

The process by which we determine limits of rational functions applies equally well to ratios containing noninteger or negative powers of x: Divide numerator and denominator by the highest power of x in the denominator and proceed from there. Find the limits. ${\displaystyle \underset{x\to \infty }{lim}\frac{x-3}{\surd 4x^2+25}}$

True or False. Every rational function has at least one vertical asymptote.