Question

Identify the asymptotes of the function F(x) = x + 2/ |x| + 3. Justify your answer analytically.

Answer 1

\[
f(x) = \frac{x+2}{|x|+3}
\]For vertical asymptote see what value(s) of x will make the denominator zero.
\(|x| + 3 = 0; ~~~~ |x| = -3\). Not possible as absolute value is always \(\ge 0\).
No vertical asymptotes.

For horizontal asymptotes, find the limit
\[\lim_ {x \rightarrow -\infty} \frac{x+2}{|x|+3} ~~~\text{and}~~~
\lim_ {x \rightarrow \infty} \frac{x+2}{|x|+3}
\]