e. \( \lim _{x \rightarrow \infty} x-\sqrt{x^{2}+x} \) f. \( \lim _{x \rightarrow-\infty} \frac{\sqrt{x^{2}-4}}{x-2} \) g. \( \lim _{x \rightarrow \infty} \frac{x^{4}+3 x}{3 x^{3}-4 x^{2}} \)

For the first limit, you can simplify \( x - \sqrt{x^2+x} \) by factoring out \( x \): rewrite it as \( x(1 - \sqrt{1 + \frac{1}{x}}) \). As \( x \) approaches infinity, the expression within the square root approaches 1, leading to the limit of 0. Regarding the second limit, when evaluating \( \frac{\sqrt{x^{2}-4}}{x-2} \) as \( x \) approaches negative infinity, you can divide the numerator and denominator by \( x \). This gives you a clearer view of the behavior of the fraction, leading to the limit being -1 because both the numerator and denominator behave like -x in the limit.