Chapter 1.3, Problem 6ES

Given that $\underset{x\to -\infty }{\lim }f\left(x\right)=7\text{ and }\underset{x\to -\infty }{\lim }g\left(x\right)=-6$ find the limits that exist. If the limit does not exist, explain why.

(a) $\underset{x\to -\infty }{\lim }\left[2f\left(x\right)-g\left(x\right)\right]$

(b) $\underset{x\to -\infty }{\lim }\left[6f\left(x\right)+7g\left(x\right)\right]$

(c) $\underset{x\to -\infty }{\lim }\left[x^2+g\left(x\right)\right]$

(d) $\underset{x\to -\infty }{\lim }\left[x^{2}g\left(x\right)\right]$

(e) $\underset{x\to -\infty }{\lim }\sqrt[3]{f\left(x\right)g\left(x\right)}$

(f) $\underset{x\to -\infty }{\lim }\frac{g\left(x\right)}{f\left(x\right)}$

(g) $\underset{x\to -\infty }{\lim }\left[f\left(x\right)\frac{g\left(x\right)}{x}\right]$

(h) $\underset{x\to -\infty }{\lim }\frac{xf\left(x\right)}{\left(2x+3\right)g\left(x\right)}$