8) $$ \lim _{x \rightarrow \infty} \frac{2 x+\cos x}{2 x} $$

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Calculate the limit $$ \lim_{xightarrow \infty} \frac{2x+\cos x}{2x} $$.
Evaluate the limit by following steps:
- step0: Evaluate the limit:
  $$\lim _{xightarrow +\infty}\left(\frac{2x+\cos\left(xight)}{2x}ight)$$
- step1: Rearrange the terms:
  $$\lim _{xightarrow +\infty}\left(\frac{\left(1+\frac{\cos\left(xight)}{2x}ight)	imes 2x}{2x}ight)$$
- step2: Reduce the fraction:
  $$\lim _{xightarrow +\infty}\left(1+\frac{\cos\left(xight)}{2x}ight)$$
- step3: Rewrite the expression:
  $$\lim _{xightarrow +\infty}\left(1ight)+\lim _{xightarrow +\infty}\left(\frac{\cos\left(xight)}{2x}ight)$$
- step4: Calculate:
  $$1+0$$
- step5: Calculate:
  $$1$$
The limit of $$ \frac{2x+\cos x}{2x} $$ as $$ x $$ approaches infinity is 1.
The limit is 1.

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