find the horizontal and vertical asymptotes of the curve. you may want comma separated lists. if an answer…

Explanation:

Step1: Find vertical asymptote

Set denominator equal to zero: (e^{x}-4 = 0), then (e^{x}=4), so (x=\ln 4).

Step2: Find horizontal asymptote

Calculate (\lim_{x\rightarrow+\infty}\frac{6e^{x}}{e^{x}-4}): Divide numerator and denominator by (e^{x}), we get (\lim_{x\rightarrow+\infty}\frac{6}{1 - 4e^{-x}}). As (x\rightarrow+\infty), (e^{-x}\rightarrow0), so (\lim_{x\rightarrow+\infty}\frac{6}{1 - 4e^{-x}}=6). Calculate (\lim_{x\rightarrow-\infty}\frac{6e^{x}}{e^{x}-4}): As (x\rightarrow-\infty), (e^{x}\rightarrow0), so (\lim_{x\rightarrow-\infty}\frac{6e^{x}}{e^{x}-4}=0).

Answer:

(x = \ln 4), (y=0,6)