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Unit Test 2 Limits and Continuity Answers

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Unit 2: Functions, Graphs, Limits, and Continuity

2a. Explain the limit of a function

  • What is the definition of the limit of a function, in your own words?
  • When x gets close to some point c, what is happening to the function value?
  • What is the difference between a one-sided limit and a two-sided limit?

The limit (L) is a way of describing what happens to a function (f(x)) near a point (c). If the limit exists, then as x gets closer to c, the function will get closer to the limit value L, even if f(c) is not defined. If the limit does not exist, the value of the function may approach more than one value, or may increase (or decrease) without bound.

To review, see The Limit of a Function and Definition of a Limit.

2b. Determine the slope of the line tangent to the graph of a function at a point

  • What is the definition of a tangent line?
  • How is the tangent line related to the secant line?
  • How do you calculate the value of the slope of the tangent line, by definition?
  • What point is both on the graph of the function, and the tangent line?

The slope of the tangent line to the graph is defined by taking the limit of the difference quotient \frac{f(x+h)-f(x)}{h} that defines the slope of a secant line. As the difference between the two points (h) gets smaller and smaller (h goes to 0), the value of the expression for the secant line will approach the slope of the tangent line. Use the expression you obtain and evaluate that expression at the point on the graph.

To review, see Tangent Lines, Velocity, and Growth.

2c. Determine the values of one- or two-sided limits for a function given by a graph

  • What is the definition of a limit, in your own words?
  • What is the difference between a jump discontinuity and a point discontinuity?
  • What conditions must be satisfied for a function to be continuous at a point?
  • When does a two-sided limit not exist?
  • What are three methods for evaluating limits?

We can describe Calculus as the study of continuous change – the limit is the basic concept that allows us to describe and analyze this change. The limit of a function describes the behavior of the function when a limit is near. A limit (L) is the value that a function (or sequence) "approaches" as the input (or index) "approaches" some value.

Limits are a fundamental concept in calculus that underpin many other concepts. For a limit to exist for a function, as x approaches a specific value c so that the difference between x and c is an arbitrarily small value, then the function value f(x) approaches some value that is arbitrarily close to the limiting value L. We can evaluate limits algebraically, numerically, or graphically. See Figure 2c.1.

Figure 2a.1.

Figure 2c.1.

Functions behave in a useful way for calculus wherever the function is continuous. There are two kinds of discontinuities for functions: jump discontinuities and point discontinuities. If the break in the function has a limit at that point (both the left- and right-handed limits exist and are the same value), then the discontinuity can be repaired by replacing a finite number of points with the values of those one-handed limits. If the one-sided limits are different, then the discontinuity is a jump discontinuity, and cannot be repaired. See Figure 2c.2.

Figure 2a.2.
Figure 2c.2.

To review, see The Limit of a Function.

2d. Use algebraic methods to determine the values of one- and two-sided limits for a function given by a formula or state that the limit does not exist

  • What are some methods for finding limits algebraically?
  • What are at least five properties of limits?
  • What are the properties of limits of composite functions?
  • What is the Squeeze Theorem?
  • What are the steps to showing that a limit does not exist?

Graphing a function or exploring a table of values to determine a limit can be cumbersome and time-consuming. When possible, it is more efficient to use the properties of limits, which is a collection of theorems for finding limits.

Knowing the properties of limits allows us to compute limits directly. We can add, subtract, multiply, and divide the limits of functions as if we were performing the operations on the functions themselves to find the limit of the result. Similarly, we can find the limit of a function raised to a power by raising the limit to that power. We can also find the limit of the root of a function by taking the root of the limit. Using these operations on limits, we can find the limits of more complex functions by finding the limits of their simpler component functions.

Review this section of the textbook which discusses properties of limits that will help you find limits algebraically, such as by using algebraic reduction or rationalization. This chapter also introduces an important theorem called the Squeeze (Squeezing) Theorem, which allows you to compare limits of functions to determine a difficult limit. See Figure 2d.1.

Figure 2b.1.a Figure 2b.1.b

Figure 2d.1.

Graphing a function or exploring a table of values to determine a limit can be cumbersome and time-consuming. When possible, it is more efficient to use the properties of limits, which is a collection of theorems for finding limits.

Knowing the properties of limits allows us to compute limits directly. We can add, subtract, multiply, and divide the limits of functions as if we were performing the operations on the functions themselves to find the limit of the result. Similarly, we can find the limit of a function raised to a power by raising the limit to that power. We can also find the limit of the root of a function by taking the root of the limit. Using these operations on limits, we can find the limits of more complex functions by finding the limits of their simpler component functions.

Properties of Limits

Let a , k , A , and B represent real numbers, and f and g be functions, such that:

 \lim_{x\rightarrow a}f(x)=A and  \lim_{x\rightarrow a}g(x)=B .

For limits that exist and are finite, the properties of limits are summarized in this table:

Constant, k \lim_{x\rightarrow a}k=k
Constant times a function \lim_{x\rightarrow a}\left [ k\cdot f(x) \right ]=k\lim_{x\rightarrow a}f(x)=kA
Sum of functions \lim_{x\rightarrow a}\left [ f(x)+g(x) \right ]=\lim_{x\rightarrow a}f(x)+\lim_{x\rightarrow a} g(x)=A+B
Difference of functions \lim_{x\rightarrow a}\left [ f(x)-g(x) \right ]=\lim_{x\rightarrow a}f(x)-\lim_{x\rightarrow a} g(x)=A-B
Product of functions \lim_{x\rightarrow a}\left [ f(x)\cdot g(x) \right ]=\lim_{x\rightarrow a}f(x)\cdot \lim_{x\rightarrow a} g(x)=A\cdot B
Quotient of functions \lim_{x\rightarrow a} \frac{f(x)}{g(x)} =\frac{\lim_{x\rightarrow a} f(x)}{\lim_{x\rightarrow a}g(x)}=\frac{A}{B},B\neq 0
Function raised to an exponent \lim_{x\rightarrow a} \left [f(x) \right ] ^n\left [ \lim_{x\rightarrow a}f(x) \right ]^n=A^n , where n is a positive integer
n th root of a function,
where n is a positive integer		 \lim_{x\rightarrow a}\sqrt[n]{f(x)}=\sqrt[n]{\lim_{x\rightarrow a}\left [ f(x) \right ]}=\sqrt[n]{A}
Polynomial function \lim_{x\rightarrow a}p(x)=p(a)

To review, see Properties of Limits.

2e. State whether a given function is continuous at a point and use the properties of continuity to find limits and values of related functions

  • What are some reasons why continuity is important for calculus?
  • What are three examples of continuous functions?
  • What are two examples of functions that are not continuous at at least one point?

A continuous function can sometimes be described as a graph that you can draw without picking up your pencil. Continuous functions play an important role in calculus because they are one of the assumptions made in many theorems. Functions can have more than one type of discontinuity: a hole (point discontinuity) or a break (jump discontinuity). See Figure 2e.1.

Most functions students encounter in calculus will be continuous on some interval (having only a finite number of discontinuities), but some special types of functions can be discontinuous everywhere.

Figure 2c.1.

Figure 2e.1.

To review, see Continuous Functions.

2f. Use the Intermediate Value Theorem to determine the number of times a function has a given value

  • What is the Intermediate Value Theorem?
  • Why does the Intermediate Value Theorem require that a function be continuous?

In mathematical analysis, the intermediate value theorem states that if f is a continuous function whose domain contains the interval [a, b], then it takes on any given value between f and f at some point within the interval.

One of the important and useful consequences of continuity is that if you have a continuous function with values f(a) and f(b), then every value between f(a) and f(b) must also be a value of the function at some point between a and b. For example, if we know that f(a) is negative and f(b) is positive, then at some point between a and b, the function must be zero. We can apply this to any value between f(a) and f(b). See Figure 2f.1.

Figure 2d.1.

Figure 2f.1.

To review, see Continuous Functions.

2g. Approximate the roots of functions using the Bisection Algorithm

  • In your own words, what is the Bisection Algorithm and how does it work?
  • What are the steps to estimating a root using the Bisection Algorithm?
  • What is an example of a function for which the Bisection Algorithm will not work?

The Bisection Algorithm is a method of finding a root or zero of a function by estimation, using the Intermediate Value Theorem. See Figure 2g.1.

First, apply the intermediate value to the whole interval to test whether there is a sign change between the endpoints. Then split the interval and check the midpoint (the point of bisection). If the function value is positive, replace the previous positive endpoint; if it is negative, replace the previous negative endpoint. Then repeat. Each time the length of the interval will get shorter and shorter until you are close enough to estimate the root. If you need more accuracy, continue the process longer.

Figure 2e.1.

Figure 2g.1.

To review, see Continuous Functions.

2h. State the epsilon-delta definition of limit

  • What is the ε-δ definition of a limit?

As we noted above, limits are a fundamental concept in calculus that underpin many other concepts. For a limit to exist for a function, as x approaches a specific value c, so that the difference between x and c is an arbitrarily small value δ, then the function value f(x) approaches some value that is arbitrarily close with a distance ε to the limiting value L. See Figure 2f.1.

Figure 2f.1.

Figure 2h.1.

To review, see Definition of a Limit.

2i. For a given epsilon, find the required delta graphically and algebraically for linear and quadratic functions

  • If you are trying to prove the limit for a linear function f(x)=mx+b, what is the relationship between ε and δ?
  • Explain the process to estimate the relationship between ε and δ if the function is not linear?
  • Is there only one correct answer to that relationship for the process described above?

To estimate the relationship between ε and δ for a linear function, you must first know the limit at the desired point, which is easy since the function is continuous everywhere. Then place the limit into the expression |f(x)-L|=ε. Factor out the expression x−c for whatever value of c you are using and replace that with δ. The two will always differ by a factor of m (the slope).

For nonlinear functions, you must first choose an interval for δ. Then estimate the value at the endpoint further away from the limiting value if you are estimating ε (use the closer value if you are estimating δ), in order to reduce your function to something linear.

To review, see Definition of a Limit.

Unit 2 Vocabulary

  • Average velocity
  • Bisection
  • Bisection algorithm
  • Break
  • Composition of functions
  • Continuity
  • Continuous function
  • Hole
  • Instantaneous velocity
  • Intermediate Value Theorem
  • Jump discontinuity
  • Limit
  • Linear function
  • Point discontinuity
  • Quadratic function
  • Rationalize
  • Root
  • Squeezing and the Squeeze Theorem

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Source: https://learn.saylor.org/mod/book/view.php?id=27460&chapterid=3320

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