- Graph each piecewise defined function.
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[latex]f(x) = \left\{ \begin{array}{lr} 2x+1 & x \leq 1 \\ -x + 2 & x > 1 \end{array} \right.[/latex]
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[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]
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- Why is the following not a function?
[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
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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.
- [latex]f(x) = x^2[/latex].
Continuousans
- 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
- 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]
Continuousans
- 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
- Your age in years as a whole number as a function of time.
Discontinuous at every birthdayans
- The height of a tennis ball as a function of time.
Continuousans
- [latex]f(x) = x^2[/latex].
- In this exercise, we will compute [latex]\lim_{x \to 0} \frac{e^x-1}{x}[/latex] using a calculator.
- 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]
- What does [latex]\lim_{x \to 0} \frac{e^x-1}{x}[/latex] seem to equal?
[latex]1[/latex]ans
- 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
- Fill in the following table of values.
- Compute the following limits using a calculator like in problem (4). In each case, sketch a graph and jot down a table of values.
- [latex]\lim_{h \to 0^+} 1 - 2^{-1/h}[/latex]
[latex]1[/latex]ans
- [latex]\lim_{h \to 3} \frac{x^2 - 9}{x - 3}[/latex]
[latex]6[/latex]ans
- [latex]\lim_{h \to 0^+} 1 - 2^{-1/h}[/latex]
- Watch the following KhanAcademy video link: One Sided Limits from Graphs
- Find the following values given the graph of [latex]f(x)[/latex] below.
- [latex]\lim_{a \to -5^+} f(a)[/latex]
[latex]1[/latex]ans
- [latex]\lim_{b \to -5^-} f(b)[/latex]
[latex]3[/latex]ans
- [latex]f(-5)[/latex]
[latex]3[/latex]ans
- [latex]\lim_{d \to -3^+} f(d)[/latex]
[latex]0[/latex]ans
- [latex]\lim_{e \to -3^-} f(e)[/latex]
[latex]0[/latex]ans
- [latex]f(-3)[/latex]
[latex]0[/latex]ans
- [latex]\lim_{g \to -1^+} f(g)[/latex]
[latex]-1[/latex]ans
- [latex]\lim_{h \to -1^-} f(h)[/latex]
[latex]5[/latex]ans
- [latex]f(-1)[/latex]
[latex]5[/latex]ans
- [latex]\lim_{j \to 1^+} f(j)[/latex]
DNE (or [latex]-\infty[/latex])ans
- [latex]\lim_{k\to 1^-} f(k)[/latex]
DNE (or [latex]-\infty[/latex])ans
- [latex]f(1)[/latex]
DNEans
- [latex]\lim_{m \to 3^+} f(m)[/latex]
[latex]-2[/latex]ans
- [latex]\lim_{n \to 3^-} f(n)[/latex]
[latex]4[/latex]ans
- [latex]f(3)[/latex]
[latex]2[/latex]ans
- [latex]\lim_{p \to 5^+} f(p)[/latex]
[latex]-1[/latex]ans
- [latex]\lim_{q \to 5^-} f(q)[/latex]
[latex]-1[/latex]ans
- [latex]f(5)[/latex]
[latex]2[/latex]ans
- [latex]\lim_{x \to \infty} f(x)[/latex]
[latex]0[/latex]ans
- [latex]\lim_{x \to -\infty} f(x)[/latex]
[latex]1[/latex]ans
- [latex]\lim_{a \to -5^+} f(a)[/latex]
- From [latex]f(x)[/latex] from Problem 7, list whether [latex]f(x)[/latex] is continuous or not continuous at the following values of [latex]x[/latex]: [latex]-5[/latex], [latex]-3[/latex], [latex]-1[/latex], [latex]1[/latex], [latex]3[/latex], [latex]5[/latex].
Only continuous at [latex]x = -3[/latex]ans
- Watch the following KhanAcademy video link:
Two-sided limit from graph - Compute (as close as you can tell from the graph) the limit or function value in each case. If the limit does not exist, explain why.
- [latex]\lim_{x \to 2} f(x)[/latex]
[latex]2[/latex]ans
- [latex]f(2)[/latex]
[latex]2[/latex]ans
- [latex]\lim_{x \to 1} f(x)[/latex]
Does not existans
- [latex]f(1)[/latex]
1ans
- [latex]\lim_{x \to 3} f(x)[/latex]
[latex]3[/latex]ans
- [latex]f(3)[/latex]
2ans
- [latex]\lim_{x \to 2} f(x)[/latex]
- Is it possible that [latex]f(2) = 3[/latex] but [latex]\lim_{x \to 2} f(x) = -3[/latex]? If so, sketch a picture of the graph of such an [latex]f(x)[/latex] If not, explain why not.
Yes, it is possible.ans
- Is it possible that [latex]\lim_{x \to 2^+}f(x) = 3[/latex] but [latex]\lim_{x \to 2} f(x) = -3?[/latex] If so, sketch a picture of the graph of such an [latex]f(x).[/latex] If not, explain why not.
- Suppose [latex]\lim_{x \to -5} f(x) = -2[/latex].
- What does [latex]f(x)[/latex] approach as [latex]x[/latex] approaches [latex]-5[/latex]?
[latex]-2[/latex]ans
- What is [latex]f(-5)?[/latex]
Unknownans
- What does [latex]x[/latex] approach if [latex]f(x)[/latex] is approaching [latex]-2[/latex]?
[latex]-5[/latex] but also perhaps other valuesans
- What does [latex]f(x)[/latex] approach as [latex]x[/latex] approaches [latex]-5[/latex]?
- Suppose [latex]\lim_{x \to 2} f(x) = -3[/latex].
- Estimate [latex]f(1.99)[/latex].
It’s probably close to [latex]-3[/latex]ans
- 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
- Estimate [latex]f(1.99)[/latex].
- For each part, sketch a graph of what [latex]f(x)[/latex] might look like. Each problem is separate and will probably require a different graph.
- [latex]\lim_{x \to 3} f(x) = 4[/latex] and [latex]f(3) = 4[/latex].
- [latex]\lim_{x \to 3} f(x)[/latex] does not exist.
- [latex]\lim_{x \to 3^+} f(x) = -2[/latex] and [latex]\lim_{x \to 3^-} f(x) = 5[/latex].
- [latex]\lim_{x \to -2} f(x) = 3[/latex] and [latex]f(-2) = 5[/latex].