Chapter 9 : Differential Calculus
901 : Lotsa Limits
by Gregory Moore
O B J E C T I V E
In this project we will examine limits from a numerical, graphical, and symbolic point of view to better understand the concept and the applicaiont of limits
S E T U P
In this project we will use the following command packages. Type and execute this line before beginning the project below. If you re-enter the worksheet for this project, be sure to re-execute this statement before jumping to any point in the worksheet.
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restart; with(plots):
Warning, the name changecoords has been redefined
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A. Review of Functions & Plots
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The format for defining a function is a little cumbersome at first, but actually very mathematical in its conception. The components of a function definition in Maple is :
name of the function
:=
independent variable(s)
crude arrow using a dash and greater than (->)
formula for the function
semi-colon to terminate the line.
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f := x -> x^4 - 4*x^2 + 5/(x + 4);
To graph a function you use the plot command with the function or its name, and a domain for x.
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plot( 9 - x^3, x = -4..4);
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plot( f(x), x = -4..4);
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#you can specifiy a range for y in this way
plot( f(x), x = -4..4, y = -4..4);
And to plot a group of functions :
This graph y=x, y=x^2, y=x^3, and y=x^4 all on same graph.
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plot( { x, x^2, x^3, x^4 }, x = -1..1);
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B. A Geometric Approach To Limits
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Now that we've review some basics, lets look at limits from a geometric point of view. The concept of limits can seem difficult, but the idea is quite simple when you see what is happening geometrically. We will construct a diagram which shows a function and points approaching from the left and right.
function definition a limit target point a the left and right endpoints of an interval near a
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f := x -> 3 + (x-2)*cos((x-2)); a := 2: left := -1: right :=5:
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display( plot( f(x), x = left..right, color = green),
plot( {[[a,0],[a,f(a)]],[[0,f(a)],[a,f(a)]] }, x = left..right,
linestyle=3,color = gold, thickness = 2),
plot([[ a - 1/n, f(a - 1/n)] $n=1..20], x = left..right,
style=point, symbol=circle, color = red),
plot( [[ a+1/n, f(a + 1/n)] $n=1..20], x = left..right,
style=point, symbol=circle, color = blue));
The green curve is the function. The yellow lines indicate where (a,f(a)) is. The blue dots indicate points on f(x) as x approaches a from the right, and the green dots indicate points on f(x) where x is converging to a from the left side. By looking at this diagram, you can guess the right limit by looking at what y value the points seem to be converging to. In a similar way, you can guess the left limit by looking at what value the red dots seem to be converging to. If the red and blue dots appear to be converging to the same value, then the limit exists and equals the value they are converging to.
Here is another example where the function is left and right limits are not the same. After re defining f(x), copy and paste the display command block above and re-execute.
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f := x -> Heaviside(x-1) - Heaviside(1-x); a := 1: left := -2: right :=4:
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display( plot( f(x), x = left..right, color = green),
plot( {[[a,0],[a,f(a)]],[[0,f(a)],[a,f(a)]] }, x = left..right,
linestyle=3,color = gold, thickness = 2),
plot([[ a - 1/n, f(a - 1/n)] $n=1..20], x = left..right,
style=point, symbol=circle, color = red),
plot( [[ a+1/n, f(a + 1/n)] $n=1..20], x = left..right,
style=point, symbol=circle, color = blue));
In this case, the function is not continuous. It has a jump at x = 1. The red points coming from the left approach -1 while the blue points approaching from the right approach +1. In this case we say that the limit is undefined since there is not a single value for the limit.
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C. A Numerical Approach To Limits
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Now that weve seen a picture of what is happening, lets take those same functions from above and examine them from a numeric point of view. We are going to take values of x that approach a from both the left and right. The x values approaching from the left will be slightly less than a, and the values from the right will be slightly larger.
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f := x -> 3 + (x-2)*cos((x-2)); a :=2:
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array( [seq([ evalf( a - 10^(-k), 15), evalf(f( a - 10^(-k)), 15)],k = 1..12)]);
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array( [seq([ evalf( a + 10^(-k), 15), evalf(f( a + 10^(-k)), 15)],k = 1..12)]);
Here is the same function we saw above when we graphed it. This first array creates a list of x values and y values which show what is happening as we approach from the left.. You can see that the x values are less than 2 but getting closer and closer to 2. The second array creates a list of x values and y values which show what happens as we approach from the right. You can see that the x values are greater than 2 but getting closer and closer to 2.
The y values are approaching the value 3 from both the left and the right. From this evidence, we can guess that the limit is 3. Lets look at another case. (All of the functions in this section were defined in the previous section and you can save some time by copying and pasting them from above rather than re-typing them.)
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f := x -> piecewise(x < 3, -5+x, x <= 6, 20-x^2, x < 10, 2*x +1.1); a:= 6:
Copy and paste the two array commands above, and re-execute them to create left and right approaches to the limit for this function. In this case the left and right limits are different. From the left, the value is 16, while from the right, the value is 13. When the left and right limits disagree, we say that the limit does not exist, since there is no single limit.
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D. A Symbolic Approach To Limits
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We're approaches limits from a geometric and a numeric point of view. Lets now address the symbolic side. Maple is able to compute the left limit, the right limit, and the limit of a function.
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restart:
f := x -> 3 + (x-2)*cos((x - 2)); a := 2:
The Limit command with a capitol "L" displays the limit in stardard mathematical notation. To find a limit just type the function and target value.
The "%" character is a reference to the most recent result processed by Maple, in this case the Limit command.
The
value
command computes the value of the limit.
For left or right limits include the option third parameter "left" or "right".
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Limit( f(x), x = a, left): % = value(%);
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Limit( f(x), x = a, right): % = value(%);
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Limit( f(x), x = a): % = value(%);
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