Limits in MapleIn this worksheet we shall compute limits with maple. We begin by examining basic strategies, go over one-sided limits, look at limits at infinity, and finally use limits to look at the limit definition of derivatives.JSFH
<Text-field style="Heading 1" layout="Heading 1">Basics</Text-field>Throughout most of this section we use the example 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1. Direct Method.The basic command for taking limits in maple is limit. In our example we would enterQyQtSSZsaW1pdEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkKiYtSSNsbkdGKDYjKiRJInhHRigiIiMiIiIsJkYvRjEhIiJGMUYzL0YvRjFGMQ==print(); # input placeholderand maple returns the limit.2. Inert Method.If we use the command with an upper case L, we obtainQyQtSSZMaW1pdEc2IjYkKiYtSSNsbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjKiRJInhHRiUiIiMiIiIsJkYvRjEhIiJGMUYzL0YvRjFGMQ==print(); # input placeholderHere our limit problem is displayed, but nothing is calculated. If we follow up with the value command, we obtain the desired result.QyQtSSZ2YWx1ZUc2IjYjSSIlR0YlIiIiprint(); # input placeholder3. Expressions TemplateThe Expressions listing on the right contains the limit command. Select it, and use the Tab-key to move through the parameters, and obtainLUkmbGltaXRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JComLUkjbG5HRiQ2IyokSSJ4R0YnIiIjIiIiLCZGLkYwISIiRjBGMi9GLkYwprint(); # input placeholder4. Define the Function or Expression FirstIf the function involved is complicated, or if the same function is used many times, it it useful to defined it first, and then apply the limit command. In our case we would do the following:QyQ+SSJmRzYiZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKiYtSSNsbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjKiQ5JCIiIyIiIiwmRjRGNiEiIkY2RjhGJUYlRiVGNg==print(); # input placeholderQyQtSSZsaW1pdEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkLUkiZkdGKDYjSSJ4R0YoL0YtIiIiRi8=print(); # input placeholderLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=More Examples: These are the three special limits on page 54 of our text.QyQtSSZsaW1pdEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkKiYtSSRzaW5HRig2I0kieEdGKCIiIkYuISIiL0YuIiIhRi8=print(); # input placeholderLUkmbGltaXRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JComLCYiIiJGKy1JJGNvc0dGJDYjSSJ4R0YnISIiRitGL0YwL0YvIiIhprint(); # input placeholderLUkmbGltaXRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JCksJiIiIkYrSSJ4R0YnRisqJEYsISIiL0YsIiIhprint(); # input placeholderLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Maple cannot perform miracles and find limits that do not exist:QyQtSSZsaW1pdEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkLUkkc2luR0YlNiMqJEkieEdGKCEiIi9GLiIiISIiIg==print(); # input placeholderQyQtSSZsaW1pdEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkKiYtSSRhYnNHRiY2I0kieEdGKCIiIkYuISIiL0YuIiIhRi8=print(); # input placeholder
<Text-field style="Heading 1" layout="Heading 1">One-Sided Limits</Text-field>For one-sided limits we need to add the direction from which the limit is taken.ExampleLet's take the piecewise defined function Ly1JImZHNiI2I0kieEdGJS1JKnBpZWNld2lzZUclKnByb3RlY3RlZEc2JjJGJyIiIywmKiRGJ0YtIiIiISIkRjAtSSM+PUdGKjYkRidGLSwmRjBGMEYnRjA= and have a look at its graph.QyQ+SSJmRzYiZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUkqcGllY2V3aXNlRyUqcHJvdGVjdGVkRzYmMjkkIiIjLCYqJEYxRjIiIiIhIiRGNTFGMkYxLCZGMUY1RjVGNUYlRiVGJUY1print(); # input placeholderQyQtSSJmRzYiNiNJInhHRiUiIiI=print(); # input placeholderQyQtSSVwbG90RzYiNiVJImZHRiU7IiIhIiIlL0koZGlzY29udEdGJUkldHJ1ZUclKnByb3RlY3RlZEciIiI=%;Clearly, we have different limits as we approach 2 from either side. Let's find themQyQtSSZsaW1pdEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYlLUkiZkdGKDYjSSJ4R0YoL0YtIiIjSSVsZWZ0R0YoIiIiprint(); # input placeholderQyQtSSZsaW1pdEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYlLUkiZkdGKDYjSSJ4R0YoL0YtIiIjSSZyaWdodEdGKCIiIg==print(); # input placeholderand the limit as such does not exitsQyQtSSZsaW1pdEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkLUkiZkdGKDYjSSJ4R0YoL0YtIiIjIiIiprint(); # input placeholderMore Examples:First we look at 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 as x approaches zero.QyQtSSZsaW1pdEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYlKiYtSSRhYnNHRiY2I0kieEdGKCIiIkYuISIiL0YuIiIhSSVsZWZ0R0YoRi8=print(); # input placeholderQyQtSSZsaW1pdEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYlKiYtSSRhYnNHRiY2I0kieEdGKCIiIkYuISIiL0YuIiIhSSZyaWdodEdGKEYvprint(); # input placeholderor, the same in clicable math,QyQtSSZsaW1pdEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYlKiYtSSRhYnNHRiY2I0kieEdGKCIiIkYuISIiL0YuIiIhSSZyaWdodEdGKEYvprint(); # input placeholderLUkmbGltaXRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JSomLUkkYWJzR0YlNiNJInhHRiciIiJGLSEiIi9GLSIiIUklbGVmdEdGJw==print(); # input placeholder
If a limit exists, the one-sided derivatives will be identical and have that value. For instance take 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at x=3.QyQ+SSJmRzYiZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKiQ5JCIiI0YlRiVGJSIiIg==print(); # input placeholderQyQtSSZsaW1pdEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYlLUkiZkdGKDYjSSJ4R0YoL0YtIiIkSSVsZWZ0R0YoIiIiprint(); # input placeholderQyQtSSZsaW1pdEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYlLUkiZkdGKDYjSSJ4R0YoL0YtIiIkSSZyaWdodEdGKCIiIg==print(); # input placeholderQyQtSSZsaW1pdEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkLUkiZkdGKDYjSSJ4R0YoL0YtIiIkIiIiprint(); # input placeholder
<Text-field style="Heading 1" layout="Heading 1">Infinity in Limits</Text-field>If you want to compute limits where x approaches infinity, just enter infinity or use the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEoJmluZmluO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDRi8= symbol from the Common Symbols list on the left.Examples:QyQtSSZsaW1pdEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkLUknYXJjdGFuR0YoNiNJInhHRigvRi1JKWluZmluaXR5R0YmIiIiprint(); # input placeholderor, equivalentlyQyQtSSZsaW1pdEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkLUknYXJjdGFuR0YlNiNJInhHRigvRi1JKWluZmluaXR5R0YmIiIiprint(); # input placeholderIf a limit does not exist because the function grows without a bound, maple will give \342\210\236 as a result.Examples:QyQtSSZsaW1pdEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkLUkjbG5HRiU2IyokSSJ4R0YoISIiL0YuIiIhIiIiprint(); # input placeholderQyQtSSZsaW1pdEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkKiYsKCokSSJ4R0YoIiIkIiIiRi0hIiciIilGL0YvLCYqJEYtIiIjRi8iIiZGLyEiIi9GLUkpaW5maW5pdHlHRiZGLw==print(); # input placeholderQyQtSSZsaW1pdEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkLUkkdGFuR0YlNiNJInhHRigvRi0sJEkjUGlHRiYjIiIiIiIjRjI=print(); # input placeholderLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=This limit is undefined, since, depending on the direction, the tangent approaches -\342\210\236 or +\342\210\236. With one-sided limits we getQyQtSSZsaW1pdEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYlLUkkdGFuR0YlNiNJInhHRigvRi0sJEkjUGlHRiYjIiIiIiIjSSVsZWZ0R0YoRjI=print(); # input placeholderQyQtSSZsaW1pdEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYlLUkkdGFuR0YlNiNJInhHRigvRi0sJEkjUGlHRiYjIiIiIiIjSSZyaWdodEdGKEYyprint(); # input placeholder
<Text-field style="Heading 1" layout="Heading 1">Limit Definition of the Derivative</Text-field>As you know, the derivative is defined as the limit 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 , and once the function f(x) is defined, we can use this command to compute the derivative.Example: Compute the derivative of the function 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. First we define the functionQyQ+SSJmRzYiZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKiY5JCIiIy1JJHNpbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjRi0iIiJGJUYlRiVGNQ==print(); # input placeholderNow compute the limitQyQtSSZsaW1pdEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkKiYsJi1JImZHRig2IywmSSJ4R0YoIiIiSSJoR0YoRjFGMS1GLTYjRjAhIiJGMUYyRjUvRjIiIiFGMQ==print(); # input placeholderExample: Compute the derivative of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYsLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInhGJ0YvRjIvRjNRJ25vcm1hbEYnRj0tSSNtb0dGJDYtUSJ+RidGPS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZULUZANi1RIj1GJ0Y9RkNGRkZIRkpGTEZORlAvRlNRLDAuMjc3Nzc3OGVtRicvRlZGZW5GPy1JJm1mcmFjR0YkNigtSSVtc3VwR0YkNiVGOi1GIzYlLUkjbW5HRiQ2JFEiMkYnRj1GL0YyLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy1GIzYnRmpuLUZANi1RIitGJ0Y9RkNGRkZIRkpGTEZORlAvRlNRLDAuMjIyMjIyMmVtRicvRlZGXHAtRmBvNiRRIjFGJ0Y9Ri9GMi8lLmxpbmV0aGlja25lc3NHRmBwLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRmVwLyUpYmV2ZWxsZWRHRkUtRiw2I1EhRidGanBGanBGPQ==. Occasionally it is useful to break the problem down into several steps, and we will do it here.QyQ+SSJmRzYiZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKiY5JCIiIywmKiRGLUYuIiIiRjFGMSEiIkYlRiVGJUYxprint(); # input placeholderQyQqJiwmLUkiZkc2IjYjLCZJInhHRiciIiJJImhHRidGK0YrLUYmNiNGKiEiIkYrRixGL0Yrprint(); # input placeholderQyQtSSlzaW1wbGlmeUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSIlR0YoIiIiprint(); # input placeholderQyQtSSZsaW1pdEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkSSIlR0YoL0kiaEdGKCIiISIiIg==print(); # input placeholderOf course, we can also go for the derivative directly, but this hides all the steps which went into the result.QyQtSSZsaW1pdEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkKiYsJi1JImZHRig2IywmSSJ4R0YoIiIiSSJoR0YoRjFGMS1GLTYjRjAhIiJGMUYyRjUvRjIiIiFGMQ==print(); # input placeholderWe can also compute the derivative using the alternate definition 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.Examples revisited:QyQ+SSJmRzYiZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKiY5JCIiIy1JJHNpbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjRi0iIiJGJUYlRiVGNQ==print(); # input placeholderQyQtSSZsaW1pdEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkKiYsJi1JImZHRig2I0kidUdGKCIiIi1GLTYjSSJ4R0YoISIiRjAsJkYvRjBGM0Y0RjQvRi9GM0Ywprint(); # input placeholderQyQ+SSJmRzYiZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKiY5JCIiIywmKiRGLUYuIiIiRjFGMSEiIkYlRiVGJUYxprint(); # input placeholderQyQsJi1JImZHNiI2I0kidUdGJiIiIi1GJTYjSSJ4R0YmISIiRik=print(); # input placeholderQyQtSSlzaW1wbGlmeUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSIlR0YoIiIiprint(); # input placeholderQyQqJkkiJUc2IiIiIiwmSSJ1R0YlRiZJInhHRiUhIiJGKkYmprint(); # input placeholderQyQtSSlzaW1wbGlmeUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSIlR0YoIiIiprint(); # input placeholderQyQtSSZsaW1pdEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkSSIlR0YoL0kidUdGKEkieEdGKCIiIg==print(); # input placeholderOr directly, asLUkmbGltaXRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JComLCYtSSJmR0YnNiNJInVHRiciIiItRiw2I0kieEdGJyEiIkYvLCZGLkYvRjJGM0YzL0YuRjI=print(); # input placeholderLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=