Final answer:
The function f(x) = (cosx)e²ˣ-1/e²ᵉ-ˣ-1+2 has no horizontal asymptote as x approaches positive infinity, but it has a horizontal asymptote at y = e as x approaches negative infinity.
Step-by-step explanation:
The function given is f(x) = (cosx)e²ˣ-1/e²ᵉ-ˣ-1+2. To find the horizontal asymptotes of the graph of f, we need to determine the behavior of f as x approaches positive infinity and negative infinity.
As x approaches positive infinity, the term e²ˣ in the numerator grows much faster than the other terms, so f(x) approaches infinity. Therefore, there is no horizontal asymptote as x approaches positive infinity.
As x approaches negative infinity, the term 1/e²ᵉ-ˣ-1 in the denominator approaches 0, while the other terms remain finite. Therefore, f(x) approaches a finite value. To find this value, we can take the limit of f(x) as x approaches negative infinity.
By applying L'Hopital's rule to the limit, we can find the value of the limit as x approaches negative infinity, which is e. Therefore, the graph of f has a horizontal asymptote at y = e as x approaches negative infinity.