Asymptote in exponentials

An asymptote is a straight line that a graph approaches more and more closely without ever touching it. Exponential graphs have a horizontal asymptote because of how they behave at the ends: at one end $y$ grows or shrinks without bound, and at the other end the curve flattens, settling towards a fixed value it never quite reaches.

For an exponential written as $y = a \cdot b^x + k$, that fixed value is the constant $k$. As the $a \cdot b^x$ part shrinks towards $0$, the whole expression approaches $k$. So the horizontal asymptote is the line $y = k$, read directly from the number added at the end.

This makes a class of questions quick: in $g(x) = 5 \cdot 2^x - 10$ the asymptote is $y = -10$; and if you are told the asymptote of $y = 2 \cdot b^x + c$ is $y = -5$, then $c = -5$.

Rational functions have asymptotes too, but of a different kind. A vertical asymptote occurs at any $x$ where the denominator equals zero while the numerator does not, because the function is undefined there. For example $f(x) = \frac{2x + 1}{x + 4}$ is undefined at $x = -4$, so $x = -4$ is a vertical asymptote.

Find the horizontal asymptote of $g(x) = 3 \cdot 4^x - 6$.
It has the form $a \cdot b^x + k$ with $k = -6$.
As $4^x \to 0$, $g(x) \to -6$, so the horizontal asymptote is $y = -6$.

The graph of $y = 3 \cdot b^x + c$ has a horizontal asymptote at $y = 2$.
The asymptote equals the constant added on.
So $c = 2$.

Find the vertical asymptote of $f(x) = \frac{x - 3}{x - 5}$.
Set the denominator to zero: $x - 5 = 0$.
So $x = 5$ is the vertical asymptote.