In general, the exponential function, \(\displaystyle y\:=\:b^x\) has a horizontal asymptote.
\(\displaystyle \;\;\)It is \(\displaystyle y\,=\,0\) (the x-axis) . . . as \(\displaystyle x\,\to\,\)-\(\displaystyle \infty\).
The graph of \(\displaystyle \,y\:=\:3^{x+3}\,+\,4\,\) is translated 3 units to the left and 4 units up.
Therefore, its horizontal asymptote is: \(\displaystyle \,y\;=\;4\)
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