Let's solve each limit one by one.

Problem Analysis

Instructions:

• Use $\text{inf}$ for $+\infty$
• Use $-\text{inf}$ for $-\infty$
• Use $\text{dne}$ if the limit does not exist

Solutions

(a) $\lim_{x \to \infty} \frac{8x^7 + 2x + 175}{13x^7 + 1}$

1. Dominant term: Since $x \to \infty$, we consider the highest power term in both numerator and denominator.
   • Numerator: $8x^7$
   • Denominator: $13x^7$
2. Simplify the expression by dividing both the numerator and denominator by $x^7$: $\lim_{x \to \infty} \frac{8x^7 + 2x + 175}{13x^7 + 1} = \lim_{x \to \infty} \frac{8 + \frac{2}{x^6} + \frac{175}{x^7}}{13 + \frac{1}{x^7}} = \frac{8}{13}$
3. Answer: $\frac{8}{13}$.

(b) $\lim_{x \to \infty} \frac{e^x}{800x^5 + \sin(x)}$

1. Dominant term: As $x \to \infty$, $e^x$ grows much faster than any polynomial term $x^5$ or the bounded term $\sin(x)$.
2. Thus, the numerator $e^x$ will tend to infinity faster than the denominator.
3. Answer: $\text{inf}$.

(c) $\lim_{x \to \infty} \frac{\sqrt{x} \log(x^{70})}{x^2}$

1. Rewrite: $\frac{\sqrt{x} \log(x^{70})}{x^2} = \frac{\sqrt{x} \cdot 70 \log(x)}{x^2} = \frac{70 \cdot x^{1/2} \log(x)}{x^2}$
2. Simplify: Rewrite $x^{1/2}$ in terms of $x^2$: $= \frac{70 \log(x)}{x^{3/2}}$
3. As $x \to \infty$, $x^{3/2}$ grows faster than $\log(x)$, so the limit approaches $0$.
4. Answer: $0$.

(d) $\lim_{x \to \infty} \left( \sqrt{x^2 + 1} - \sqrt{x^2 - 1} \right)$

1. Multiply by the conjugate: $\left( \sqrt{x^2 + 1} - \sqrt{x^2 - 1} \right) \cdot \frac{\sqrt{x^2 + 1} + \sqrt{x^2 - 1}}{\sqrt{x^2 + 1} + \sqrt{x^2 - 1}}$ gives $= \frac{(x^2 + 1) - (x^2 - 1)}{\sqrt{x^2 + 1} + \sqrt{x^2 - 1}} = \frac{2}{\sqrt{x^2 + 1} + \sqrt{x^2 - 1}}$
2. Simplify: As $x \to \infty$, $= \frac{2}{x\sqrt{1 + \frac{1}{x^2}} + x\sqrt{1 - \frac{1}{x^2}}} = \frac{2}{2x} = \frac{1}{x}$
3. As $x \to \infty$, $\frac{1}{x} \to 0$.
4. Answer: $0$.

Final Answers:

1. (a) $\frac{8}{13}$
2. (b) $\text{inf}$
3. (c) $0$
4. (d) $0$

Would you like any of these steps explained further?

Relative Questions

1. How do we determine the dominant terms in polynomial expressions when calculating limits at infinity?
2. Why is $e^x$ dominant over polynomial expressions as $x \to \infty$?
3. When is it beneficial to use the conjugate method in limit problems?
4. How does logarithmic growth compare to polynomial and exponential growth as $x \to \infty$?
5. What does it mean for a limit to "not exist," and how does that differ from approaching $\infty$ or $-\infty$?

Tip

When evaluating limits as $x \to \infty$, focus on the highest power term in each part of the expression, as lower power terms become negligible.