Why am I not getting the right answer for this limit problem?

Question

The problem:

problem

"Let

f(x) = 3x + sqrt(9x^2 + 2x + 1)

Calculate

a) lim(x→∞) = f(x)

b) lim(x→-∞) = f(x)."

My calculations / what I have tried:

part 1

part 2

part 3

part 4

The answer key to this problem:

answer key

"a) DNE! [“∞ + ∞”]

b) -1/3 (expand with the conjugate)"

How my math book solves a similar problem:

part 1

part 2

My questions to you:

• What am I doing wrong? Why won’t I get the right answers? Have I made an error along the way?

• Why is [“∞ + ∞”] only of relevance in question a)? How do I come to the conclusion “∞ + ∞” for a)?

• And ultimately, how would you solve this problem?

Please, help me understand. I have read pdfs and watched multiple videos on these specific type of problems and they all seem to have employed a similar method to mine but my calculations won’t lead me to the right answer it seems. What am I doing wrong?

Answer 1

If the limit on an algebraic expression $\lim_{x\rightarrow x_0}f(x)$ is not defined for $f(x_0)$, it is customary to find another expression $g(x)=f(x)$ on wich the limit has a defined value. Because of this, several procedures lead to different limits. The key is to search for the ones that give you a reasonable answer. In your exercise, you’ll have a better luck if you do not use the absolute value function $|\cdot|$. To see this, check these calculations: $$3x+\sqrt{9x^2+2x+1}=\frac{-(2x+1)}{3x-\sqrt{9x^2+2x+1}}$$ On the limit, we get $$\lim_{x\rightarrow - \infty} \frac{-(2x+1)}{3x-\sqrt{9x^2+2x+1}} = \frac{-(2(-\infty)+1)}{3(-\infty)-\sqrt{9(-\infty)^2+2(-\infty)+1}} = \frac{2\infty}{-3\infty-\sqrt{9\infty^2}} = \frac{2\infty}{-3\infty-3\infty} = \frac{2\infty}{-6\infty}=-\frac{1}{3}$$