Asymptotes
An asymptote is a line that a curve approaches, as it heads towards infinity.
We need to find if the given function has a defined trend when x heads towards infinity or find the limit when x increases or decreases without limits
[tex]\lim _{x\to-oo\text{ }}3^{\mleft\{2x+1\mright\}}+3[/tex]The exponent tends to zero when x tends to minus infinity, thus:
[tex]\lim _{x\to-oo\text{ }}3^{\{2x+1\}}+3=0+3=3[/tex]The line y=3 is a horizontal asymptote.
When x tens to plus infinity, the given function tends to infinity also, thus the limit:
[tex]\lim _{x\to+oo\text{ }}3^{\{2x+1\}}+3=\infty[/tex]Does not exist.
There is only one asymptote with the equation y=3
