[Limits at infinity examples] Example. The graph of a function y = f(x) is shown below. Compute lim f(x) and lim f(x).

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1 [Limits at infinity eamples] Eample. The graph of a function y = f() is shown below. Compute f() and f(). y -8 As you go to the far right, the graph approaches y =, so f() =. As you go to the far left, the graph approaches y = 8, so f() = 8. Eample. The graph of a function y = f() is shown below. Compute f() and f(). y As you go to the far right, the graph goes up without bound, so f() =. As you go to the far left, the graph approaches the -ais (that is, y = 0), so f() = 0. In the problems that follow, you should give eact answers, and do the problems algebraically. 7 Eample. Compute = 0.

2 Many of these its will use these rules: If k is a number and n is a positive number, If If k is a number and n is a positive integer, k n = 0. k n = 0. The second formula holds if n is a positive number but not an integer, as long as n is defined for negative values of. As an eample where the second formula doesn t work, / is undefined. Eample. Compute 05. Remember that: (a) An even power of a negative number is positive. (For eample, () = 6.) (b) An odd power of a negative number is negative. (For eample, () 3 = 8.) As gets large, 5 gets large. Since, and since the fifth power of a negative number is negative, 05 =. Eample. Compute e8. Note that e 8 = e 8. As, I have 8, so e8, and hence e8 = Thus, e8 Eample. Compute cos. Here s the graph of cos: cos(*) cos does not approach a single number,, or as (as you move along the graph to the far right). Hence, cos is undefined.

3 Eample. Compute The highest power of on the top and on the bottom is. I divide the top and bottom of the fraction by : = = = = = = 98 5 = 6 5. Eample. Compute powers: To see what the highest powers are, write the fractions as negative = Notice that 3 > 0 and 3 > 5. So the highest power on the top is 3 = and the highest 3 power on the bottom is 3 = 3. I will clear these powers by multiplying the top and the bottom by 3 : = 5 = = 7 7+ = = 7. Eample. Compute The biggest power on the top is 3 and the biggest power on the bottom is. I ll divide the top and bottom of the fraction by : = 3 3 = = 3 =.

4 Here is where the answer came from. The top is approaching The bottom is approaching The whole fraction is approaching 3 =. ( ) 6+0 =. 03 = = 3 3 = 3 =. +7 Eample. Compute The biggest power on the top is and the biggest power on the bottom is 3. I ll divide the top and bottom of the fraction by 3 : = = = 0 5 = = = = 5 3 = 5 = 0. Eample. Compute. 6+ Before starting, note that, so is negative. Look at the fraction: The top () is negative, and the bottom ( 6+ ) is positive (since always means positive square root. So the whole fraction is (negative), which means that it is negative. (positive) All of this means that the answer can t be a positive number. Now I ll do the algebra. The highest power on the top is. The highest power on the bottom is, but it s inside the square root, so it is effectively = (maybe). So it looks like I should divide the top and the bottom by : 6+ = 6+ = 6+ = ( ) (6+ ) = (6+ ) = 6+ = = =. What happened after the third equality, when the moved inside the square root?

5 First, when you pull something out of a square root, it gets square rooted: 8 (a+b) = 8 a+b = 8 a+b. So when you push something into a square root, it gets squared: 8 a+b = 8 (a+b). Well, this is what happens with 8. But in this problem,, so is negative, so is negative. This bring us to the second issue: The negative sign. Look at 6+, which is what you have before the moves inside. It is a negative number ( ) times a positive number ( 6+ ), which is negative. ( ) ( ) Hence, I need to write (6+ ), because without the negative sign I d have ( +5), which is positive. All together, when I move a negative term ( ) inside a square root, it gets squared and leaves a negative sign outside. There would be no negative sign if the it had. c 08 by Bruce Ikenaga 5