Evaluate \( \lim _{x \rightarrow-\infty}\left(6+\frac{8}{x}\right) \) and \( \lim _{x \rightarrow \infty}\left(6+\frac{8}{x}\right) \). Determine the horizontal asymptotes of \( f(x)=\left(6+\frac{8}{x}\right) \), if any. \( \lim _{x \rightarrow-\infty}\left(6+\frac{8}{x}\right)= \) Number \( \lim _{x \rightarrow \infty}\left(6+\frac{8}{x}\right)= \) Number The field below accepts a list of numbers or formulas separated by semicolons \( ( \) e.g. \( 2 ; 4 ; 6 \) or \( x+1 ; x-1) \). The order of the list does not matter. If there is no horizontal asymptote, enter NA. Horizontal Asymptote: \( y=\square \) ⿴

Evaluate the limit by following steps: - step0: Evaluate the limit: \(\lim _{x\rightarrow +\infty}\left(6+\frac{8}{x}\right)\) - step1: Rewrite the expression: \(\lim _{x\rightarrow +\infty}\left(6\right)+\lim _{x\rightarrow +\infty}\left(\frac{8}{x}\right)\) - step2: Calculate: \(6+0\) - step3: Calculate: \(6\) Calculate the limit \( \lim_{x\rightarrow -\infty} \left(6+\frac{8}{x}\right) \). Evaluate the limit by following steps: - step0: Evaluate the limit: \(\lim _{x\rightarrow -\infty}\left(6+\frac{8}{x}\right)\) - step1: Rewrite the expression: \(\lim _{x\rightarrow -\infty}\left(6\right)+\lim _{x\rightarrow -\infty}\left(\frac{8}{x}\right)\) - step2: Calculate: \(6+0\) - step3: Calculate: \(6\) Let's evaluate the limits step by step: 1. **Evaluate \( \lim _{x \rightarrow -\infty}\left(6+\frac{8}{x}\right) \)**: - As \( x \) approaches \(-\infty\), the term \( \frac{8}{x} \) approaches \( 0 \) because the denominator becomes very large in magnitude (negative). - Therefore, the limit simplifies to: \[ \lim _{x \rightarrow -\infty}\left(6+\frac{8}{x}\right) = 6 + 0 = 6 \] 2. **Evaluate \( \lim _{x \rightarrow \infty}\left(6+\frac{8}{x}\right) \)**: - As \( x \) approaches \( \infty\), the term \( \frac{8}{x} \) also approaches \( 0 \) because the denominator becomes very large. - Thus, the limit simplifies to: \[ \lim _{x \rightarrow \infty}\left(6+\frac{8}{x}\right) = 6 + 0 = 6 \] Both limits yield the same result: - \( \lim _{x \rightarrow -\infty}\left(6+\frac{8}{x}\right) = 6 \) - \( \lim _{x \rightarrow \infty}\left(6+\frac{8}{x}\right) = 6 \) ### Horizontal Asymptote Since both limits approach the same value, the function \( f(x) = 6 + \frac{8}{x} \) has a horizontal asymptote at \( y = 6 \). ### Final Answers - \( \lim _{x \rightarrow -\infty}\left(6+\frac{8}{x}\right) = 6 \) - \( \lim _{x \rightarrow \infty}\left(6+\frac{8}{x}\right) = 6 \) - Horizontal Asymptote: \( y = 6 \)