evaluate the following limit. give an exact answer, which may be in terms of ( a ). otherwise, ente ( -infty…

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)}= )

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)}= )

Answer

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

Step3: Simplify the expression

[ \begin{align*} \lim_{x\rightarrow\infty}\frac{e^{3x}}{\frac{e^{3x}-e^{-3x}}{2}}&=\lim_{x\rightarrow\infty}\frac{2e^{3x}}{e^{3x}-e^{-3x}}\ &=\lim_{x\rightarrow\infty}\frac{2}{1 - e^{-6x}} \end{align*} ]

Step4: Evaluate the limit as (x\rightarrow\infty)

As (x\rightarrow\infty), (e^{-6x}=\frac{1}{e^{6x}}\rightarrow0). So, (\lim_{x\rightarrow\infty}\frac{2}{1 - e^{-6x}}=\frac{2}{1-0}).

Answer:

(2)