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Limits at Infinity Study Guide (page 2)

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Updated on Oct 1, 2011

Going to Infinity

The whole concept of "going to infinity" might be a bit confusing. This really means "going toward infinity," because infinity is not something that a real number can reach. So don't wander off pondering the one number that is bigger than all the rest (unless you enjoy that). Just know that "going to infinity" means using really big numbers, and that "going to negative infinity" means using really big negative numbers.

Example 1

Evaluate .

Solution 1

The most powerful part of the numerator is – x2, and in the denominator is x3 Thus:

.

This goes to zero because the numerator is clearly outclassed by the more powerful denominator. Also, as x gets really big, gets really close to zero. For example, when x = 1,000, then = = 0.001.

Example 2

Evaluate .

Solution 2

Here, the numerator and denominator are evenly matched, with each having x2 as its highest power of x.

The limit is formed by the coefficients of the most powerful parts: 3 in the numerator and –8 in the denominator.

Example 3

Evaluate .

Solution 3

Here,

As x goes to infinity, x8 also gets really large, but the negative in the – 5 reverses this and makes – 5x8 approach negative infinity.

The infinite limits of ex and In(x) can be seen from their graphs in Figure 13.4.

Figure 13.4

In general, as x goes to infinity, ex is more powerful than x raised to any number. The natural logarithm, however, goes to infinity slower than just about anything else. It may look as though y = In( x) is beginning to level out into a horizontal asymptote, but actually, it will eventually surpass any height as it slowly goes up to infinity.

In more complicated situations, we use L'Hôpital's rule. This states that if the numerator and denominator both go to infinity (positive or negative), then the limit remains the same after taking the derivative of the top and the bottom.

 

L'Hôpital's Rule

If the numerator and denominator both go to infinity (positive or negative), the limit remains the same after taking the derivative of the top and bottom, OR:

Example 4

Evaluate .

Solution 4

Since and , we can use L'Hôpital's Rule.

Note: The little H over the equals sign indicates that L'Hôpital's Rule as been used at that point. Examples like this demonstrate how ln(x) goes to infinity even slower than x does.

Example 5

Evaluate .

Solution 5

Here, and , so we use L'Hôpital's Rule.

Notice that we don't use the Quotient Rule, because we take the derivative of the numerator and denominator separately. Here, we need to use L'Hôpital's Rule several more times:

This example shows how ex is more powerful than x3. If the denominator had an x100, we'd have to use L'Hôpital's Rule 100 times, but in the end, the ex would take everything to infinity.

Example 6

Evaluate .

Solution 6

This is a trick question! The limit is not infinite, so we can't use L'Hôpital's Rule. The function ex is only powerful when x goes to positive infinity, Instead, we use the old "plug in" method (or common sense).

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