1. [latex]f(x) = \left\{ \begin{array}{lr} 2x+1 & x \leq 1 \\ -x + 2 & x > 1 \end{array} \right.[/latex]

2. [latex]g(x) = \left\{ \begin{array}{lr} \frac{1}{2}x + 3 & x \leq -4 \\ -\frac{1}{2}x - 1 & x > -4 \end{array} \right.[/latex]

[latex]h(x) = \left\{ \begin{array}{lr} \frac{2}{3}x & x \leq 1 \\ \frac{4}{3}x - 5 & x \geq 1 \end{array} \right.[/latex]

Fundamentals

If [latex]\lim_{x \to a} f(x) = f(a)[/latex], then we say that [latex]f(x)[/latex] is continuous at point [latex]a[/latex]. In general, there is a simpler way to think about continuity: if a section of a graph of a function can be drawn without lifting your pencil, then that part of the function is continuous. If you need to lift your pencil at some point, that point is called a discontinuity. If the whole function is continuous everywhere, then the function itself is called continuous.

For each function below, label it as continuous or not continuous. If it is not continuous, list at least one discontinuity.

1. [latex]f(x) = x^2[/latex].
   
   Continuous

   ans
2. The piecewise linear function [latex]g(x)[/latex] defined by

   [latex]g(x) = \left\{ \begin{array}{lr} -x & x \leq 2 \\ 2x & x > 2 \end{array} \right.[/latex]

   Discontinuous at [latex]x = 2[/latex].

   ans
3. The piecewise linear function [latex]g(x)[/latex] defined by

   [latex]g(x) = \left\{ \begin{array}{lr} x+1 & x \leq 3 \\ 2x-2 & x > 3 \end{array} \right.[/latex]

   Continuous

   ans
4. The piecewise linear function [latex]g(x)[/latex] defined by

   [latex]g(x) = \left\{ \begin{array}{lr} 3x+11 & x \leq -5 \\ 0 & x > -5 \end{array} \right.[/latex]

   Discontinuity at [latex]x = -5[/latex]

   ans
5. Your age in years as a whole number as a function of time.
   
   Discontinuous at every birthday

   ans
6. The height of a tennis ball as a function of time.
   
   Continuous

   ans

1. Fill in the following table of values.

   [latex]\begin{array}{|r|r|} \hline x & \frac{e^x-1}{x} \\ \hline -0.1 & \\ -0.01 & \\ -0.001 & \\ 0.001 & \\ 0.01 & \\ 0.1 & \\ \hline \end{array}[/latex]

2. What does [latex]\lim_{x \to 0} \frac{e^x-1}{x}[/latex] seem to equal?
   
   [latex]1[/latex]

   ans
3. Use a graphing calculator or computer to graph [latex]y = \frac{e^x-1}{x}[/latex]. Looking at the graph, what does it look like [latex]y[/latex] is equal to when [latex]x = 0[/latex]?
   
   Loooooks like [latex]1[/latex].

   ans

1. [latex]\lim_{h \to 0^+} 1 - 2^{-1/h}[/latex]
   
   [latex]1[/latex]

   ans
2. [latex]\lim_{h \to 3} \frac{x^2 - 9}{x - 3}[/latex]
   
   [latex]6[/latex]

   ans

1. [latex]\lim_{a \to -5^+} f(a)[/latex]
   
   [latex]1[/latex]

   ans
2. [latex]\lim_{b \to -5^-} f(b)[/latex]
   
   [latex]3[/latex]

   ans
3. [latex]f(-5)[/latex]
   
   [latex]3[/latex]

   ans
4. [latex]\lim_{d \to -3^+} f(d)[/latex]
   
   [latex]0[/latex]

   ans
5. [latex]\lim_{e \to -3^-} f(e)[/latex]
   
   [latex]0[/latex]

   ans
6. [latex]f(-3)[/latex]
   
   [latex]0[/latex]

   ans
7. [latex]\lim_{g \to -1^+} f(g)[/latex]
   
   [latex]-1[/latex]

   ans
8. [latex]\lim_{h \to -1^-} f(h)[/latex]
   
   [latex]5[/latex]

   ans
9. [latex]f(-1)[/latex]
   
   [latex]5[/latex]

   ans
10. [latex]\lim_{j \to 1^+} f(j)[/latex]
    
    DNE (or [latex]-\infty[/latex])

    ans
11. [latex]\lim_{k\to 1^-} f(k)[/latex]
    
    DNE (or [latex]-\infty[/latex])

    ans
12. [latex]f(1)[/latex]
    
    DNE

    ans
13. [latex]\lim_{m \to 3^+} f(m)[/latex]
    
    [latex]-2[/latex]

    ans
14. [latex]\lim_{n \to 3^-} f(n)[/latex]
    
    [latex]4[/latex]

    ans
15. [latex]f(3)[/latex]
    
    [latex]2[/latex]

    ans
16. [latex]\lim_{p \to 5^+} f(p)[/latex]
    
    [latex]-1[/latex]

    ans
17. [latex]\lim_{q \to 5^-} f(q)[/latex]
    
    [latex]-1[/latex]

    ans
18. [latex]f(5)[/latex]
    
    [latex]2[/latex]

    ans
19. [latex]\lim_{x \to \infty} f(x)[/latex]
    
    [latex]0[/latex]

    ans
20. [latex]\lim_{x \to -\infty} f(x)[/latex]
    
    [latex]1[/latex]

    ans

1. [latex]\lim_{x \to 2} f(x)[/latex]
   
   [latex]2[/latex]

   ans
2. [latex]f(2)[/latex]
   
   [latex]2[/latex]

   ans
3. [latex]\lim_{x \to 1} f(x)[/latex]
   
   Does not exist

   ans
4. [latex]f(1)[/latex]
   
   1

   ans
5. [latex]\lim_{x \to 3} f(x)[/latex]
   
   [latex]3[/latex]

   ans
6. [latex]f(3)[/latex]
   
   2

   ans

1. What does [latex]f(x)[/latex] approach as [latex]x[/latex] approaches [latex]-5[/latex]?
   
   [latex]-2[/latex]

   ans
2. What is [latex]f(-5)?[/latex]
   
   Unknown

   ans
3. What does [latex]x[/latex] approach if [latex]f(x)[/latex] is approaching [latex]-2[/latex]?
   
   [latex]-5[/latex] but also perhaps other values

   ans

1. Estimate [latex]f(1.99)[/latex].
   
   It’s probably close to [latex]-3[/latex]

   ans
2. Is is possible [latex]f(1.99) = -42[/latex]? Why or why not?
   
   This is possible but unlikely. There is nothing in the definition of limit that says [latex]f(1.99)[/latex] can’t equal [latex]-42[/latex] if the limit as [latex]x \to 2[/latex] is [latex]-3[/latex].

   ans

1. [latex]\lim_{x \to 3} f(x) = 4[/latex] and [latex]f(3) = 4[/latex].
2. [latex]\lim_{x \to 3} f(x)[/latex] does not exist.
3. [latex]\lim_{x \to 3^+} f(x) = -2[/latex] and [latex]\lim_{x \to 3^-} f(x) = 5[/latex].
4. [latex]\lim_{x \to -2} f(x) = 3[/latex] and [latex]f(-2) = 5[/latex].