Limits Involving Infinity
Seeing is believing! So we begin with a geometric approach.
Example 1: Consider the following graph of a function y=f(x) :
This function can be defined as the piecewise function
We can use this function definition to directly compute the limits illustrated in the above graph:
See if you can understand why each of the steps above is true. Compare this with the above graph, and also with the following table. In this table, you can also easily discern the patterns, and see what the limits are!
x
f(x)
–10000
1.9999
–1000
1.999
–100
1.99
–10
1.9
–1
1
–0.1
–8
–0.01
–98
–0.001
–998
–0.0001
–9998
–0.00001
–99998
0.00001
100001
0.0001
10001
0.001
1001
0.01
101
0.1
11
1
2
10
1.1
100
1.01
1000
1.001
10000
1.0001
Intuitively,
says that the bigger x gets, the closer f(x) gets to L;
says that the smaller x gets, the closer f(x) gets to L;
says that the closer x gets to A, the bigger f(x) gets;
says that the closer x gets to A, the smaller f(x) gets.
The above should give you an intuitive understanding of limits involving infinity. This should suffice for now, as it did historically for over 150 years after Newton and Leibniz first put calculus on the (cognitive) map. However, mathematicians began finding bizarre exceptions to these intuitive notions, and ultimately had to make the basic definitions much more precise (though somewhat counterintuitive, at least to beginning students). Thus calculus was finally given a solid foundation to build upon. To see a rigorous (formal) definition of limits involving infinity, click here.
When the limit of a function as x goes to positive or negative infinity is a fixed value, L, as in the graph above, the line y=L is called a horizontal asymptote of the function (these are the yellow lines in the graph). When a one-sided limit of a function, as x goes to A, equals positive or negative infinity, then x=A is called a vertical asymptote of the function (the y-axis is a vertical asymptote in the above graph).
Horizontal Asymptotes of Rational Functions
We’ll illustrate one commonly used technique in the next two examples. Note that we’ll then learn a much easier method of handling these types of problems.
Example 2:
We begin by multiplying the numerator and denominator by the factor 1/x5:
This limit approaches infinity because the denominator goes to 0 while the numerator goes to 1, and 0 divides 1 infinitely many times. Also, this ratio approaches positive infinity because 1 is the biggest term in the numerator (as x gets large), and 1/x is the biggest term in the denominator (as x gets large), and the ratio of two positive quantities is positive.
Example 3:
We begin by multiplying the numerator and denominator by the factor 1/x5:
This limit approaches infinity because the denominator goes to 0 while the numerator goes to 1, and 0 divides 1 infinitely many times. Also, this ratio approaches positive infinity because 1 is the biggest term in the numerator (as x gets large), and 1/x is the biggest term in the denominator (as x gets large), and the ration of two positive quantities is positive.
If you’re ready for an easier approach to these kinds of problems, here it is! This rule is very simple, and applies whenever the numerator and denominator consist of sums of powers of x (not just integer powers, ANY real exponents will do). The principle behind this technique is simply this: The term with the largest exponent will dominate all the others as x becomes large. To see why, just compare x with x2, as x goes to infinity… When x=10, x2 is 10 times larger than x. When x=1000000, x2 is 1000000 times larger than x. You can see that x2 is rapidly leaving x in the dust! This means that you only need to look at the highest power of x in the numerator and the highest power of x in the denominator, as all the rest fades to insignificance (as x gets very large). Next we’ll try this new idea on those last two examples.
Note that exponential functions (with positive exponents) grow faster than any polynomial function.
Example 2 (revisited):
Throw away all but the most significant terms in the numerator and in the denominator. The resulting ratio will have the same limit!
Example 3 (revisited):
This technique applies even if the exponents aren’t integers!
Example 4:
Rewrite the radicals using fractional exponents, then throw away all but the highest degree term in the numerator, and in the denominator:
Example 5:
Rewrite these radicals using fractional exponents, then throw away all but the highest degree term in the numerator, and in the denominator:
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