Find the limit. (If an answer does not exist, enter DNE.) $$ \lim _{x \rightarrow \infty} \arctan \left(e^{x}\right) $$

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Calculate the limit \[ \lim _{x \rightarrow \infty} \arctan \left(e^{x}\right) $$.
Evaluate the limit by following steps:
- step0: Evaluate the limit:
  $$\lim _{x\rightarrow +\infty}\left(\arctan\left(e^{x}\right)\right)$$
- step1: Rewrite the expression:
  $$\arctan\left(\lim _{x\rightarrow +\infty}\left(e^{x}\right)\right)$$
- step2: Calculate:
  $$\arctan\left(+\infty\right)$$
- step3: Calculate the trigonometric value:
  $$\frac{\pi }{2}$$
The limit of $$ \lim _{x \rightarrow \infty} \arctan \left(e^{x}\right) $$ is $$ \frac{\pi}{2} $$.
The limit is $$ \frac{\pi}{2} $$.

Find the limit. (If an answer does not exist, enter DNE.)

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