1
Give a polynomial \(f(x)\) with the property that both \(\displaystyle\lim_{x \rightarrow\infty} f(x)\) and \(\displaystyle\lim_{x \rightarrow -\infty} f(x)\) are (finite) real numbers.
Subsection 1.5.2 Exercises
Exercises — Stage 1
2
Give a polynomial \(f(x)\) that satisfies \(\displaystyle\lim_{x \rightarrow\infty} f(x) \neq \displaystyle\lim_{x \rightarrow -\infty} f(x)\text{.}\)
Exercises — Stage 2
3
Evaluate \(\displaystyle\lim_{x \rightarrow \infty} 2^{-x}\)
4
Evaluate \(\displaystyle\lim_{x \rightarrow \infty} 2^x\)
5
Evaluate \(\displaystyle\lim_{x \rightarrow -\infty} 2^x\)
6
Evaluate \(\displaystyle\lim_{x \rightarrow -\infty} \cos x\)
7
Evaluate \(\displaystyle\lim_{x \rightarrow\infty}x-3x^5+100x^2\text{.}\)
8
Evaluate \(\displaystyle\lim_{x \rightarrow\infty} \dfrac{\sqrt{3x^8+7x^4}+10}{x^4-2x^2+1}\text{.}\)
9 (✳)
\(\lim\limits_{x\rightarrow \infty} \left[\sqrt{x^2+5x}-\sqrt{x^2-x}\right]\)
10 (✳)
Evaluate \(\displaystyle \lim_{x\to -\infty} \frac{3x}{\sqrt{4x^2+x}-2x}\text{.}\)
11 (✳)
Evaluate \(\lim\limits_{x\rightarrow -\infty}\dfrac{1-x-x^2}{2x^2-7}\text{.}\)
12 (✳)
Evaluate \(\lim\limits_{x\rightarrow\infty}\big(\sqrt{x^2+x}-x\big)\)
13 (✳)
Evaluate \(\displaystyle \lim_{x\to +\infty} \frac{5x^2-3x+1}{3x^2 +x+7}.\)
14 (✳)
Evaluate \(\displaystyle \lim_{x\to +\infty} \frac{ \sqrt{4\,x + 2}}{3\,x+4}\text{.}\)
15 (✳)
Evaluate \(\displaystyle \lim_{x\to +\infty} \frac{4x^3+x}{7x^3 + x^2 - 2}\text{.}\)
16
Evaluate \(\displaystyle\lim_{x \rightarrow -\infty}\dfrac{\sqrt[3]{x^2+x}-\sqrt[4]{x^4+5}}{x+1}\)
17 (✳)
Evaluate \(\displaystyle\lim_{x\rightarrow +\infty} \frac{5x^2+10}{3x^3 +2x^2+x}.\)
18
Evaluate \(\displaystyle\lim_{x \rightarrow -\infty}\frac{x+1}{\sqrt{x^2}}\text{.}\)
19
Evaluate \(\displaystyle\lim_{x \rightarrow \infty}\frac{x+1}{\sqrt{x^2}}\)
20 (✳)
Find the limit \(\displaystyle \lim_{x\to -\infty} \sin\left( \frac{\pi}{2} \frac{|x|}{x}\right) + \frac{1}{x}\text{.}\)
21 (✳)
Evaluate \(\displaystyle \lim_{x\to -\infty} \frac{3x+5}{\sqrt{x^2+5}-x}\text{.}\)
22 (✳)
Evaluate \(\displaystyle\lim_{x\rightarrow -\infty} \frac{5x+7}{\sqrt{4x^2+15}-x}\)
23
Evaluate \(\displaystyle\lim_{x \rightarrow -\infty}\dfrac{3x^7+x^5-15}{4x^2+32x}\text{.}\)
24 (✳)
Evaluate \(\ds\lim_{n \to \infty}\left(\sqrt{n^2+5n}-n\right)\text{.}\)
25
Evaluate \(\ds\lim_{a \to 0^+}\dfrac{a^2-\frac{1}{a}}{a-1}\text{.}\)
26
Evaluate \(\ds\lim_{x \to 3}\dfrac{2x+8}{\frac{1}{x-3}+\frac{1}{x^2-9}}\text{.}\)
Exercises — Stage 3
27
Give a rational function \(f(x)\) with the properties that \(\displaystyle\lim_{x \rightarrow\infty} f(x) \neq \displaystyle\lim_{x \rightarrow -\infty} f(x)\text{,}\) and both limits are (finite) real numbers.
28
Suppose the concentration of a substance in your body \(t\) hours after injection is given by some formula \(c(t)\text{,}\) and \(\displaystyle\lim_{t \rightarrow \infty} c(t) \neq 0\text{.}\) What kind of substance might have been injected?