Problem Set: The Precise Definition of a Limit

In the following exercises (1-4), write the appropriate [latex]\varepsilon[/latex]–[latex]\delta[/latex] definition for each of the given statements.

1. [latex]\underset{x\to a}{\lim}f(x)=N[/latex]

2. [latex]\underset{t\to b}{\lim}g(t)=M[/latex]

3. [latex]\underset{x\to c}{\lim}h(x)=L[/latex]

4. [latex]\underset{x\to a}{\lim}\phi(x)=A[/latex]

The following graph of the function [latex]f[/latex] satisfies [latex]\underset{x\to 2}{\lim}f(x)=2[/latex]. In the following exercises (5-6), determine a value of [latex]\delta >0[/latex] that satisfies each statement.

A function drawn in quadrant one for x > 0. It is an increasing concave up function, with points approximately (0,0), (1, .5), (2,2), and (3,4).

5. If [latex]0<|x-2|<\delta[/latex], then [latex]|f(x)-2|<1[/latex].

6. If [latex]0<|x-2|<\delta[/latex], then [latex]|f(x)-2|<0.5[/latex].

The following graph of the function [latex]f[/latex] satisfies [latex]\underset{x\to 3}{\lim}f(x)=-1[/latex]. In the following exercises (7-8), determine a value of [latex]\delta >0[/latex] that satisfies each statement.

A graph of a decreasing linear function, with points (0,2), (1,1), (2,0), (3,-1), (4,-2), and so on for x >= 0.

7. If [latex]0<|x-3|<\delta[/latex], then [latex]|f(x)+1|<1[/latex].

8. If [latex]0<|x-3|<\delta[/latex], then [latex]|f(x)+1|<2[/latex].

The following graph of the function [latex]f[/latex] satisfies [latex]\underset{x\to 3}{\lim}f(x)=2[/latex]. In the following exercises (9-10), for each value of [latex]\varepsilon[/latex], find a value of [latex]\delta >0[/latex] such that the precise definition of limit holds true.

A graph of an increasing linear function intersecting the x axis at about (2.25, 0) and going through the points (3,2) and, approximately, (1,-5) and (4,5).

9. [latex]\varepsilon =1.5[/latex]

10. [latex]\varepsilon =3[/latex]

In the following exercises (11-12), use a graphing calculator to find a number [latex]\delta[/latex] such that the statements hold true.

11. [T] [latex]|\sin (2x)-\frac{1}{2}|<0.1[/latex], whenever [latex]|x-\frac{\pi}{12}|<\delta[/latex]

12. [T] [latex]|\sqrt{x-4}-2|<0.1[/latex], whenever [latex]|x-8|<\delta[/latex]

In the following exercises (13-17), use the precise definition of limit to prove the given limits.

13. [latex]\underset{x\to 2}{\lim}(5x+8)=18[/latex]

14. [latex]\underset{x\to 3}{\lim}\dfrac{x^2-9}{x-3}=6[/latex]

15. [latex]\underset{x\to 2}{\lim}\dfrac{2x^2-3x-2}{x-2}=5[/latex]

16. [latex]\underset{x\to 0}{\lim}x^4=0[/latex]

17. [latex]\underset{x\to 2}{\lim}(x^2+2x)=8[/latex]

In the following exercises (18-20), use the precise definition of limit to prove the given one-sided limits.

18. [latex]\underset{x\to 5^-}{\lim}\sqrt{5-x}=0[/latex]

19. [latex]\underset{x\to 0^+}{\lim}f(x)=-2[/latex], where [latex]f(x)=\begin{cases} 8x-3 & \text{ if } \, x<0 \\ 4x-2 & \text{ if } \, x \ge 0 \end{cases}[/latex]

20. [latex]\underset{x\to 1^-}{\lim}f(x)=3[/latex], where [latex]f(x)=\begin{cases} 5x-2 & \text{ if } \, x < 1 \\ 7x-1 & \text{ if } x \ge 1 \end{cases}[/latex]

In the following exercises (21-23), use the precise definition of limit to prove the given infinite limits.

21. [latex]\underset{x\to 0}{\lim}\dfrac{1}{x^2}=\infty[/latex]

22. [latex]\underset{x\to -1}{\lim}\dfrac{3}{(x+1)^2}=\infty[/latex]

23. [latex]\underset{x\to 2}{\lim}-\dfrac{1}{(x-2)^2}=−\infty[/latex]

24. An engineer is using a machine to cut a flat square of Aerogel of area 144 cm2. If there is a maximum error tolerance in the area of 8 cm2, how accurately must the engineer cut on the side, assuming all sides have the same length? How do these numbers relate to [latex]\delta, \, \varepsilon, \, a[/latex], and [latex]L[/latex]?

25. Use the precise definition of limit to prove that the following limit does not exist: [latex]\underset{x\to 1}{\lim}\dfrac{|x-1|}{x-1}[/latex]

26. Using precise definitions of limits, prove that [latex]\underset{x\to 0}{\lim}f(x)[/latex] does not exist, given that [latex]f(x)[/latex] is the ceiling function.

27. Using precise definitions of limits, prove that [latex]\underset{x\to 0}{\lim}f(x)[/latex] does not exist: [latex]f(x)=\begin{cases} 1 & \text{ if } \, x \, \text{is rational} \\ 0 & \text{ if } \, x \, \text{is irrational} \end{cases}[/latex]

28. Using precise definitions of limits, determine [latex]\underset{x\to 0}{\lim}f(x)[/latex] for [latex]f(x)=\begin{cases} x & \text{ if } \, x \, \text{is rational} \\ 0 & \text{ if } \, x \, \text{is irrational} \end{cases}[/latex]

29. Using the function from the previous exercise, use the precise definition of limits to show that [latex]\underset{x\to a}{\lim}f(x)[/latex] does not exist for [latex]a\ne 0[/latex].

For the following exercises (30-32), suppose that [latex]\underset{x\to a}{\lim}f(x)=L[/latex] and [latex]\underset{x\to a}{\lim}g(x)=M[/latex] both exist. Use the precise definition of limits to prove the following limit laws:

30. [latex]\underset{x\to a}{\lim}(f(x)-g(x))=L-M[/latex]

31. [latex]\underset{x\to a}{\lim}[cf(x)]=cL[/latex] for any real constant [latex]c[/latex]

32. [latex]\underset{x\to a}{\lim}[f(x)g(x)]=LM[/latex].