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Subsection 3.7.4 Exercises

Exercises — Stage 1

In Questions 3.7.4.1 to 3.7.4.3, you are asked to give pairs of functions that combine to make indeterminate forms. Remember that an indeterminate form is indeterminate precisely because its limit can take on a number of values.

1

Give two functions \(f(x)\) and \(g(x)\) with the following properties:

  1. \(\ds\lim_{x \to \infty} f(x)=\infty\)
  2. \(\ds\lim_{x \to \infty} g(x)=\infty\)
  3. \(\ds\lim_{x \to \infty} \dfrac{f(x)}{g(x)}=2.5\)
2

Give two functions \(f(x)\) and \(g(x)\) with the following properties:

  1. \(\ds\lim_{x \to \infty} f(x)=\infty\)
  2. \(\ds\lim_{x \to \infty} g(x)=\infty\)
  3. \(\ds\lim_{x \to \infty} \dfrac{f(x)}{g(x)}=0\)
3

Give two functions \(f(x)\) and \(g(x)\) with the following properties:

  1. \(\ds\lim_{x \to \infty} f(x)=1\)
  2. \(\ds\lim_{x \to \infty} g(x)=\infty\)
  3. \(\ds\lim_{x \to \infty} [f(x)]^{g(x)}=5\)

Exercises — Stage 2

4 (✳)

Evaluate \(\lim\limits_{x\rightarrow 1}\dfrac{x^3-e^{x-1}}{\sin(\pi x)}\text{.}\)

5 (✳)

Evaluate \(\lim\limits_{x\rightarrow 0+}\dfrac{\log x}{x}\text{.}\) (Remember: in these notes, \(\log\) means logarithm base \(e\text{.}\))

6 (✳)

Evaluate \(\lim\limits_{x\rightarrow\infty}(\log x)^2e^{-x}\text{.}\)

7 (✳)

Evaluate \(\lim\limits_{x\rightarrow\infty}x^2e^{-x}\text{.}\)

8 (✳)

Evaluate \(\lim\limits_{x\rightarrow 0}\dfrac{x-x\cos x}{x-\sin x}\text{.}\)

9

Evaluate \(\ds\lim_{x \to 0}\dfrac{\sqrt{x^6+4x^4}}{x^2\cos x}\text{.}\)

10 (✳)

Evaluate \(\lim\limits_{x\rightarrow\infty}\dfrac{(\log x)^2}{x}\text{.}\)

11 (✳)

Evaluate \(\lim\limits_{x\rightarrow0}\dfrac{1-\cos x}{\sin^2 x}\text{.}\)

12

Evaluate \(\ds\lim_{x \to 0}\dfrac{x}{\sec x}\text{.}\)

13

Evaluate \(\ds\lim_{x\to0}\dfrac{\csc x\cdot \tan x\cdot (x^2+5)}{e^x}\text{.}\)

14

Evaluate \(\ds\lim_{x \to \infty}\sqrt{2x^2+1}-\sqrt{x^2+x}\text{.}\)

15 (✳)

Evaluate \(\lim\limits_{x\rightarrow 0}\dfrac{\sin(x^3+3x^2)}{\sin^2x}\text{.}\)

16 (✳)

Evaluate \(\lim\limits_{x\rightarrow1}\dfrac{\log(x^3)}{x^2-1}\text{.}\)

17 (✳)

Evaluate \(\lim\limits_{x\rightarrow 0}\dfrac{e^{-1/x^2}}{x^4}\text{.}\)

18 (✳)

Evaluate \(\lim\limits_{x\rightarrow 0} \dfrac{xe^x}{\tan (3x)}\text{.}\)

19

Evaluate \(\lim\limits_{x \to 0}\sqrt[x^2]{\sin^2 x}\text{.}\)

20

Evaluate \(\lim\limits_{x \to 0}\sqrt[x^2]{\cos x}\text{.}\)

21

Evaluate \(\ds\lim_{x \to 0^+} e^{x \log x}\text{.}\)

22

Evaluate \(\ds\lim_{x \rightarrow 0} \left[-\log(x^2)\right]^x\text{.}\)

23 (✳)

Find \(c\) so that \(\lim\limits_{x\rightarrow 0} \dfrac{1+cx-\cos x}{ e^{x^2}-1}\) exists.

24 (✳)

Evaluate \(\lim\limits_{x\rightarrow 0}\dfrac{e^{k\sin(x^2)}-(1+2x^2)}{x^4}\text{,}\) where \(k\) is a constant.

Exercises — Stage 3

25

Suppose an algorithm, given an input with with \(n\) variables, will terminate in at most \(S(n)=5n^4-13n^3-4n+\log (n)\) steps. A researcher writes that the algorithm will terminate in roughly at most \(A(n)=5n^4\) steps. Show that the percentage error involved in using \(A(n)\) instead of \(S(n)\) tends to zero as \(n\) gets very large. What happens to the absolute error?

Remark: this is a very common kind of approximation. When people deal with functions that give very large numbers, often they don't care about the exact large number--they only want a ballpark. So, a complicated function might be replaced by an easier function that doesn't give a large relative error. If you would like to know more about the ways people describe functions that give very large numbers, you can read about “Big O Notation” in Section 3.6.3 of the CLP2 textbook.