Determine whether the function below has any horizontal asymptotes. $f ( x ) = \frac { e ^ { x } - e ^ { - x } } { 2 }$
A)horizontal asymptotes: y = 0 and y = 2
B)no horizontal asymptotes
C)horizontal asymptotes: y = 1
D)horizontal asymptotes: y = 1 and y = 3
E)horizontal asymptotes: y = 3

Question

Quizplus · Accepted Answer

The function $f(x) = \frac{e^x - e^{-x}}{2}$ is known as the hyperbolic sine function, denoted as sinh(x). As $x$ approaches infinity, $e^x$ grows much faster than $e^{-x}$ diminishes, leading to the function approaching infinity without leveling off at a constant value. Similarly, as $x$ approaches negative infinity, $e^{-x}$ grows, and $e^x$ diminishes, causing the function to approach negative infinity. Therefore, it does not have any horizontal asymptotes.

Determine Whether the Function Below Has Any Horizontal Asymptotes f(x)=ex−e−x2f ( x ) = \frac { e ^ { x } - e ^ { - x } } { 2 }f(x)=2ex−e−x​

Determine Whether the Function Below Has Any Horizontal Asymptotes $f ( x ) = \frac { e ^ { x } - e ^ { - x } } { 2 }$