Chapter 1.3, Problem 5ES

Given that $\underset{x\to +\infty }{\lim }f\left(x\right)=3,\underset{x\to +\infty }{\lim }g\left(x\right)=-5,\underset{x\to +\infty }{\lim }h\left(x\right)=0$ find the limits that exist. If the limit does not exist, explain why.

(a) $\underset{x\to +\infty }{\lim }\left[f\left(x\right)+3g\left(x\right)\right]$

(b) $\underset{x\to +\infty }{\lim }\left[h\left(x\right)-4g\left(x\right)+1\right]$

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

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

(e) $\underset{x\to +\infty }{\lim }\sqrt[3]{5+f\left(x\right)}$

(f) $\underset{x\to +\infty }{\lim }\frac{3}{g\left(x\right)}$

(g) $\underset{x\to +\infty }{\lim }\frac{3h\left(x\right)+4}{x^2}$

(h) $\underset{x\to +\infty }{\lim }\frac{6f\left(x\right)}{5f\left(x\right)+3g\left(x\right)}$