Question

1. find the horizontal asymptote(s) of

(i) ( y=\frac{3 x^{9}+5 x+2}{4 x^{9}-7 x^{4}+3 x} ) (ii) ( y=\frac{4 e^{2 x}+3 e^{x}+5}{left(1-2 e^{x}
ight)left(e^{x}+3
ight)} ) (iii) ( y=\frac{sqrt{4 x^{2}+7}}{5 x-3} ).

Explanation:

Step1: Divide numerator and denominator by \(x^9\)

For \(y = \frac{3x^{9}+5x + 2}{4x^{9}-7x^{4}+3x}\), divide numerator and denominator by \(x^9\):

$$
LATEXBLOCK0
$$

Step2: Evaluate the limit

As \(x
ightarrow\pm\infty\), \(\frac{1}{x^{n}}
ightarrow0\) for \(n>0\). So,

$$
LATEXBLOCK1
$$

Step3: Simplify the second function

For \(y=\frac{4e^{2x}+3e^{x}+5}{(1 - 2e^{x})(e^{x}+3)}=\frac{4e^{2x}+3e^{x}+5}{e^{2x}+3e^{x}-2e^{x}-6}=\frac{4e^{2x}+3e^{x}+5}{e^{2x}+e^{x}-6}\)
Divide numerator and denominator by \(e^{2x}\):

$$
LATEXBLOCK2
$$

$$
LATEXBLOCK3
$$

Step4: Simplify the third function

For \(y = \frac{\sqrt{4x^{2}+7}}{5x-3}\), when \(x
ightarrow+\infty\):

$$
LATEXBLOCK4
$$

When \(x
ightarrow-\infty\):

$$
LATEXBLOCK5
$$

Answer:

(i) \(y = \frac{3}{4}\)
(ii) \(y = 4\) and \(y=-\frac{5}{6}\)
(iii) \(y=\frac{2}{5}\) and \(y =-\frac{2}{5}\)