Question

evaluate the following limit. give an exact answer, which may be in terms of ( a ). otherwise, ente ( -infty ) or ( infty ) if the limit is infinite, or enter dne if the limit does not exist in another way. ( lim _{x \to infty} \frac{e^{3 x}}{sinh (3 x)}= )

Explanation:

Step1: Recall the definition of hyperbolic sine

The hyperbolic sine function is defined as \(\sinh(t)=\frac{e^{t}-e^{-t}}{2}\). So, \(\sinh(3x)=\frac{e^{3x}-e^{-3x}}{2}\).

Step2: Substitute the definition into the limit

We have \(\lim_{x
ightarrow\infty}\frac{e^{3x}}{\sinh(3x)}=\lim_{x
ightarrow\infty}\frac{e^{3x}}{\frac{e^{3x}-e^{-3x}}{2}}\).

Step3: Simplify the expression

$$
LATEXBLOCK0
$$

Step4: Evaluate the limit as \(x

ightarrow\infty\)
As \(x
ightarrow\infty\), \(e^{-6x}=\frac{1}{e^{6x}}
ightarrow0\). So, \(\lim_{x
ightarrow\infty}\frac{2}{1 - e^{-6x}}=\frac{2}{1-0}\).

Answer:

\(2\)