91Ó°ÊÓ

We will first break the given function into three parts: 1. \(\lim _{x \rightarrow -\infty} 5 \) 2. \(\lim _{x \rightarrow -\infty} \frac{100}{x} \) 3. \(\lim _{x \rightarrow -\infty} \frac{\sin ^{4} x^{3}}{x^{2}}\) Now, we will find the limit of each part.

1. \(\lim _{x \rightarrow -\infty} 5 \): This limit represents a horizontal asymptote, which is a constant value, so the limit will be equal to the constant value regardless of x, so the limit equals 5. 2. \(\lim _{x \rightarrow -\infty} \frac{100}{x} \): As x approaches negative infinity, the fraction will approach 0 because the denominator is growing incredibly large, so the limit equals 0. 3. \(\lim _{x \rightarrow -\infty} \frac{\sin ^{4}{x^{3}}}{x^{2}}\): Since the sine function's values are bounded between -1 and 1, the numerator squared will be always between 0 and 1. As x approaches negative infinity, the denominator, \(x^{2}\), will always be positive and approach positive infinity. Thus, the fraction will approach 0 as x approaches negative infinity. Therefore, the limit equals 0.

Now that we have the limits of each part, we can combine them by sum: $$\lim _{x \rightarrow-\infty}\left(5+\frac{100}{x}+\frac{\sin ^{4} x^{3}}{x^{2}} \right) = 5 + 0 + 0$$ So, the final answer is: $$\lim _{x \rightarrow-\infty}\left(5+\frac{100}{x}+\frac{\sin ^{4} x^{3}}{x^{2}} \right) = 5$$