Try dividing both the numerator and denominator of $f(x)$ by $5^x$. This will create terms with $5^{-x}$, which become very small as $x$ increases without bound.

Once you know the $y$-value that $f(x)$ approaches, look at $g(x)=6+\dfrac{3}{5^x-3}$ and $h(x)=6\Bigl(1+\dfrac{1}{2(5^{x}-3)}\Bigr)$. In each case, what happens to the fractional part as $x$ gets very large, and which number remains?

The question asks where the asymptote’s $y$-value appears as a constant or coefficient. After you find the limiting value, check whether it shows up as a standalone number or a multiplying factor in I, in II, or in both.

In Desmos, enter the function f(x) = (6*5^x - 15) / (5^x - 3) in an expression line so you can see its graph.

Pan or zoom the graph to the right (large positive $x$ values). Watch how the graph levels off and approaches a horizontal line; move the cursor along the curve at large $x$ (for example, $x=5,10,15$) and read off the $y$-values to see what number they are approaching.

Once you see which $y$-value the function is approaching, type a horizontal line with that value (for example, y = 6) into a new expression line and check that the curve of $f(x)$ gets closer and closer to this line as $x$ increases.

Add g(x) = 6 + 3/(5^x - 3) and h(x) = 6*(1 + 1/(2*(5^x - 3))). Notice that they overlap the graph of $f(x)$ (for allowed $x$-values) and that in each equation the same $y$-value you observed for the horizontal asymptote appears explicitly: as a constant term in $g(x)$ and as a coefficient in $h(x)$. Use this to decide which statements are correct.

A horizontal asymptote is the $y$-value that a function approaches as $x$ becomes very large (for this question, as $x$ increases without bound). So we need to find $\lim_{x\to\infty} f(x)$ and then see which of the given equivalent forms of $f$ shows that $y$-value clearly as a constant or coefficient.

Start with

Divide the numerator and denominator by $5^x$:

As $x\to\infty$, $5^{-x}\to 0$, so the small terms with $5^{-x}$ vanish:

Therefore, the horizontal asymptote of the graph of $y=f(x)$ is the line $y=6$. This is the $y$-coordinate we are looking for in the equivalent equations.

Equation I is

As $x$ becomes large, the denominator $5^x-3$ becomes very large, so the fraction $\dfrac{3}{5^x-3}$ gets very close to 0. That means

In this form, the horizontal asymptote’s $y$-value appears directly as the constant term 6 in the expression $6+\text{(small fraction)}$.

Equation II is

Again, as $x$ becomes large, the denominator $2(5^x-3)$ becomes very large, so the fraction $\dfrac{1}{2(5^x-3)}$ goes to 0, and the parentheses approach 1:

Here, the horizontal asymptote’s $y$-value appears as the coefficient 6 multiplying a factor that approaches 1.

So both I and II display the $y$-coordinate of the horizontal asymptote as a constant or coefficient. The correct answer choice is I and II.