2.6E: Exercises

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Reading Questions

What is the key difference in the precise definition of an infinite limit at a finite point (e.g., \( \displaystyle \lim_{x \to a} f(x) = \infty\)) compared to a finite limit at a finite point, in terms of the "challenge" value (\(N\) vs. \(\epsilon\)) and the condition on \(f(x)\)?
In the precise definition \( \displaystyle \lim_{x \to a} f(x) = \infty\), what does \(N \gg 0\) signify?
How is the definition of a vertical asymptote related to infinite limits at finite numbers?
When defining a finite limit at infinity (e.g., \( \displaystyle \lim_{x \to \infty} f(x) = L\)), what replaces the \(\delta\)-neighborhood around \(a\)? What is the condition on \(x\)?
State the precise definition for \( \displaystyle \lim_{x \to \infty} f(x) = L\).
State the precise definition for \( \displaystyle \lim_{x \to -\infty} f(x) = L\).
How is the definition of a horizontal asymptote related to finite limits at infinity?
In the precise definition of an infinite limit at infinity (e.g., \( \displaystyle \lim_{x \to \infty} f(x) = \infty\)), what two quantities are we relating using \(N\) and \(M\)?
State the precise definition for \( \displaystyle \lim_{x \to \infty} f(x) = \infty\).
State the precise definition for \( \displaystyle \lim_{x \to -\infty} f(x) = -\infty\).
What does the theorem regarding \( \displaystyle \lim_{x \to \infty} \frac{1}{x^p}\) state for \(p>0\)?
When proving \( \displaystyle \lim_{x \to -2^+} \ln(x+2) = -\infty\), why is a one-sided limit considered?

Homework

In exercises 1 - 10, use the precise definitions to prove the given limits.

1) \(\displaystyle \lim_{x \to 0}\frac{1}{x^2}= \infty\)

2) \(\displaystyle \lim_{x \to −1}\frac{3}{(x+1)^2}= \infty\)

Answer: Let \( \delta =\sqrt{\frac{3}{N}}\). If \(0 <|x+1| <\sqrt{\frac{3}{N}}\), then \(f(x)=\frac{3}{(x+1)^2} >N\).

3) \(\displaystyle \lim_{x \to 2} −\frac{1}{(x −2)^2}= − \infty\)

4) \(\displaystyle \lim_{x \to \pi^+} \ln{(x −\pi)}= − \infty\)

5) \(\displaystyle \lim_{x \to \infty} \frac{2}{x^6} = 0\)

6) \(\displaystyle \lim_{x \to -\infty} -\frac{5}{x^3} = 0\)

7) \(\displaystyle \lim_{x \to \infty} a x^6 = \infty \), where \( a \gt 0 \)

8) \(\displaystyle \lim_{x \to -\infty} (-10 + 2 x^5) = -\infty \)

9) \(\displaystyle \lim_{x \to -\infty} e^{-x} = \infty \)

10) \(\displaystyle \lim_{x \to -\infty} e^{x} = 0 \)

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