Limits of Functions

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some large value of x for which 1/x 2 is smaller than r Therefore, as x grows without bound, 1/x 2 approaches 0, telling us that Lim 1/x 2 = 0

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Still another example Again, let x be a positive real-number variable This time, let s evaluate

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Lim 1/x 2

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Suppose that we start out with x at some positive real number for which the function is defined and then decrease x, letting it get arbitrarily close to 0 but always remaining positive As we decrease the value of x, the value of 1/x 2 remains positive and increases If we choose some large positive real number s, no matter how gigantic, we can always find some small, positive value of x for which 1/x 2 is larger than s As x becomes arbitrarily small positively, 1/x 2 grows without bound, so

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Lim 1/x 2

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Are you confused

It s easy to get mixed up by the meanings of negative direction and positive direction, and how these relate to the notions of left hand and right hand These terms are based on the assumption that we re talking about the horizontal axis in a graph, and that this axis represents the independent variable In most graphs of this type, the value of the independent variable gets more negative as we move to the left, and it gets more positive as we move to the right As we travel along the horizontal axis, we might be in positive territory the whole time; we might be in negative territory the whole time; we might cross over from the negative side to the positive side or vice versa Whenever we come toward a point from the left, we approach from the negative direction, even if that point corresponds to something like x = 567 Whenever we come toward a point from the right, we approach from the positive direction, even if the point is at x = 53,535 The location of the point doesn t matter The important consideration is the direction from which we approach the point

Here s a challenge!

Consider the base-10 logarithm function (symbolized log10) Sketch a graph of the function f (x) = log10 x for values of x from 01 to 10, and for values of f from 1 to 1 Then determine Lim log10 x

x 5

Solution

Figure 19-5 is a graph of the function f (x) = log10 x for values of x from 01 to 10, and for values of f from 1 to 1 The function varies smoothly throughout this span If we start at values of x a little smaller than 5 and work our way toward 5, the value of f approaches log10 5 Therefore, Lim log10 x = log10 5

x 5

Sequences, Series, and Limits

f (x)

We close in on this point

from the negative direction

0 5 10

Common logarithm function f (x) = log 10 x

Figure 19-5

An example of the limit of a function as we approach a point from the negative direction

Memorable Limits of Series

Certain limits of series are found often in calculus and analysis If you plan to go on to Calculus Know-It-All after finishing this book, you re certain to see the three examples that follow!

An example Imagine an infinite series where we take a positive integer i and then divide it by the square of another positive integer n Symbolically, we write this as

i /n

i =1

When we expand this series out, we write it as 1/n 2 + 2/n 2 + 3/n 2 + + n/n 2 + which simplifies to (1 + 2 + 3 + + n + )/n 2

Memorable Limits of Series

Suppose that we let n grow endlessly larger, increasing the number of terms in the series Let s consider Lim (1 + 2 + 3 + + n)/n 2

As things work out, this limit is equal to 1/2 Therefore Lim

i /n

i =1

= 1/2

Another example Now imagine an infinite series where we square a positive integer i and then divide it by the cube of another positive integer n Symbolically, we write this as