Question

What is the equation of the line that represents the horizontal asymptote of the function f(x)=25,000(1+0.025)^(x)?

Answer 1

The answer is below

Explanation:

The horizontal asymptote of a function f(x) is gotten by finding the limit as x ⇒ ∞ or x ⇒ -∞. If the limit gives you a finite value, then your asymptote is at that point.

`\lim_(x \to \infty) f(x)=A\\\\or\\\\ \lim_(x \to -\infty) f(x)=A\\\\where\ A\ is\ a\ finite\ value.\\\\Given\ that \ f(x) =25000(1+0.025)^x\\\\ \lim_(x \to \infty) f(x)= \lim_(x \to \infty) [25000(1+0.025)^x]= \lim_(x \to \infty) [25000(1.025)^x]\\=25000 \lim_(x \to \infty) [(1.025)^x]=25000(\infty)=\infty\\\\ \lim_(x \to -\infty) f(x)= \lim_(x \to -\infty) [25000(1+0.025)^x]= \lim_(x \to -\infty) [25000(1.025)^x]\\=25000 \lim_(x \to -\infty) [(1.025)^x]=25000(0)=0\\\\`

`Since\ \lim_(x \to -\infty) f(x)=0\ is\ a\ finite\ value,hence:\\\\Hence\ the\ horizontal\ asymtotes\ is\ at\ y=0`