The Dragon's Tail Function is defined as .

Although these methods for finding limits are easy to use, in some cases they do not work. It takes practice to know which methods work and when they don't. Here is one of my examples.

Possible Problems with Graphical and Numerical Estimates

For further practice with answers click here: Limits

e)

d)

c)

b)

a)

Find the following limits symbolically.

In the calculus toolbar you will find the commands to find limits symbolically.

Symbolic Methods

Numerically investigate the limit .

Step 6) Since these two estimates appear to be approaching the same limit,

L = 1.099, then by the theorem, the value 1.099 is an estimate for the two-sided limit. Compare this with:

Step 5) Estimate the left-hand limit

Step 4) Estimate the right-hand limit

This step is necessary for a function to accept vectors as an input for calculation, if you use a range variable you do not need to vectorize the function, the problem with range variables is they are harder to use in this case since you usually have too many values.

Step 3) Define and vectorize the function .

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and

Find

h) Given

g) Given find

when x = 2.

f)

for

e)

d)

c)

a)

b)

The format of your worksheet should clearly identify the graphical, numerical and symbolic methods for finding the following limits. The graphical solution should include 1) the plot of the limit point if possible, and 2) a statement of the estimated limit. For the numerical solution, numerically determine the limit if it exists 1) from above, 2) from below and 3) two-sided. (Except for problems d and h) For each problem, give a concluding statement that gives the limit, and explain any variations between the various methods.

Find the following limits, if they exist, graphically, numerically and symbolically.

Assignment: Due (1 week)

Why do you suppose that this function is called the Dragon's Tail?

c) Explain the results you found in a) and b).

to estimate the limit.

b) Use the sequence

a) Graphically estimate the limit .

Estimate of L = 0.16666

L

(always reset your range variable)

Step 3) Plot and label the limit point L on your graph.

Graphing in the connect-the-line mode creates a bogus spike. Why?

Step 2) Graph (in point mode ), then use the trace utility to estimate the coordinates of the limit point, L.

Step 1) Write the expression in function form.

Solution:

Estimate this limit by graphing:

We should know by now that even though the graph appears to approach 1, we do not know that the limit is 1! Nevertheless, 1 is a likely candidate. Graphically, we estimate:

As x nears 0, the value of appears to get close to 1.

Look at the graph of f(x) .

Estimate ,

that is as x approaches the target value of 0, what number does the function values appear to approach?

Estimating the limit of a function as x approaches a specified value is often easily accomplished by graphing the function over the interval that contains the target value and using visual analysis.

Finding Limits Graphically

In order to evaluate this limit, it is customary at this time to study techniques for finding limits. After completing this lab you will be familiar with graphical, numerical and symbolic methods for evaluating limits.

One of the central goals of the differential calculus is illustrated geometrically as finding the "slope" of a curve at a given point. What is the slope of a curve at a point? In the Zoom lab we explored this concept. In the Secant worksheet we developed an algebraic representation for the zoom, and saw how the slope of a set of secant lines could be used to determine the slope of a curve at a point. This amounts to finding the limit:

Why Find Limits?

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Limits of a Function

Graphical, Numerical and Symbolic Methods

(x period l)

Step 2) Create a sequence of numbers xl that converge to 0,

from the left-side.

(x period r)

Step 1) Create a sequence of numbers xr that converge to 0,

from the right-side. We will use these to approximate the right-sided limit.

We use the theorem to numerically determine this limit.

Numerically investigate the limit .

and

if and only if it has left-hand and right-hand limits and these limits equal ,

Theorem: A function has a limit as approaches

Another way to estimate the limit of a function is by numerical methods. Numerical estimates of the limit are found by inputting into the limit expression a sequence of numbers that converge to the target value . It is necessary to use number sequences that converge from both sides to the target value , unless you are estimating a one-sided limit.

Estimate Limits Numerically

Change a to -1, 0, 1 and 2

K(x) is a Piecewise-defined function stringing Boolean conditions using “and” and “or.”

Estimate the limit