evaluate the following limits. if needed, enter inf for ∞ and -inf for -∞. (a) lim x→∞ √(11 + 10x²) / (9 +…

Explanation:

Step1: Divide numerator and denominator by x

When (x\to\infty), for (x>0), (\sqrt{11 + 10x^{2}}=x\sqrt{\frac{11}{x^{2}}+10}). Divide (\frac{\sqrt{11 + 10x^{2}}}{9 + 3x}) by (x): (\frac{\frac{\sqrt{11 + 10x^{2}}}{x}}{\frac{9 + 3x}{x}}=\frac{\sqrt{\frac{11}{x^{2}}+10}}{\frac{9}{x}+3}).

Step2: Evaluate the limit as (x\to\infty)

As (x\to\infty), (\lim_{x\to\infty}\frac{11}{x^{2}} = 0) and (\lim_{x\to\infty}\frac{9}{x}=0). Then (\lim_{x\to\infty}\frac{\sqrt{\frac{11}{x^{2}}+10}}{\frac{9}{x}+3}=\frac{\sqrt{0 + 10}}{0+3}=\frac{\sqrt{10}}{3}).

Step3: Consider (x\to-\infty)

When (x\to-\infty), (\sqrt{11 + 10x^{2}}=-x\sqrt{\frac{11}{x^{2}}+10}) (since (x<0)). Divide (\frac{\sqrt{11 + 10x^{2}}}{9 + 3x}) by (x): (\frac{\frac{\sqrt{11 + 10x^{2}}}{x}}{\frac{9 + 3x}{x}}=\frac{-\sqrt{\frac{11}{x^{2}}+10}}{\frac{9}{x}+3}).

Step4: Evaluate the limit as (x\to-\infty)

As (x\to-\infty), (\lim_{x\to-\infty}\frac{11}{x^{2}} = 0) and (\lim_{x\to-\infty}\frac{9}{x}=0). Then (\lim_{x\to-\infty}\frac{-\sqrt{\frac{11}{x^{2}}+10}}{\frac{9}{x}+3}=-\frac{\sqrt{0 + 10}}{0+3}=-\frac{\sqrt{10}}{3}).

Answer:

(a) (\frac{\sqrt{10}}{3}) (b) (-\frac{\sqrt{10}}{3})