Algebraic+Methods+for+Evaluating+Limits

**A L G E B R A I C  M E T H O D <span style="color: #ff7a7a; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif; font-size: 360%;">S <span style="color: #000080; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif; font-size: 360%;"> F <span style="color: #3232cd; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif; font-size: 360%;">O <span style="color: #b8b8db; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif; font-size: 360%;">R <span style="color: #6e0202; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif; font-size: 360%;"> E <span style="color: #d00101; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif; font-size: 360%;">V <span style="color: #ff4d4d; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif; font-size: 360%;">A <span style="color: #f90606; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif; font-size: 360%;">L <span style="color: #ffc2c2; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif; font-size: 360%;">U <span style="color: #ff00ff; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif; font-size: 360%;">A <span style="color: #ab3bab; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif; font-size: 360%;">T <span style="color: #974e97; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif; font-size: 360%;">I <span style="color: #ba5a5a; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif; font-size: 360%;">N <span style="color: #931b93; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif; font-size: 360%;">G <span style="color: #000000; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif; font-size: 360%;"> L <span style="color: #404040; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif; font-size: 360%;">I <span style="color: #808080; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif; font-size: 360%;">M <span style="color: #cfcfcf; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif; font-size: 360%;">I <span style="color: #d9d9d9; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif; font-size: 360%;">TS <span style="color: #000000; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif; font-size: 360%;">!! ****<span style="color: #050605; font-family: 'Palatino Linotype','Book Antiqua',Palatino,serif; font-size: 210%;"> The most basic.... ** Usually, when evaluating limits, the most simple method is ** Direct Substitution ** or replacing the variable ‘x’ in the equation with the c value that x is approaching For example.... When you substitute 2 into the equation, you get 3/1, or 3.

Here's another, worked out, example: lim x à 2 (3x2-4) = ? Substitute 2 for x: lim x à 2 (3x2-4) à lim x à 2 (3(2)2-4) à lim x à 2 (3x2-4) = <span style="-moz-background-clip: -moz-initial; -moz-background-inline-policy: -moz-initial; -moz-background-origin: -moz-initial; background: yellow none repeat scroll 0% 0%;">8 <span style="color: #dd0303; font-family: Impact,Charcoal,sans-serif; font-size: 250%;"> But it isn't always that simple!!! (Almost never actually.) In the world of calculus, often limit equations end up in the unfortunate indeterminate (0/0) form. In no case is this okay! When this happens, you need to take affirmative action.


 * So you've got this limit and you're like oh no, indeterminate form, how do I solve this? Don't panic, you can factor it!!...sometimes**

For example, say you've got this crazy function like

When you substitute 5 for x, you end up with (25-25)/(25+5-30) or (0/0). Now you have reached the intermediate form. This means there is a hole or removable discontinuity at x =5. All you have to do is factor it. So then you get From there you can simplify it to

Then you can substitute 5 for x and you get 10/11.

Success!! Let's do another one.

Here, we have a case of the tricky absolute value sign. Don't let it scare you! In this case, use left and right limits to solve.

For example, the absolute value of X-2 as X approaches 2 from the left is <0. Therefore, we can rewrite the limit as...

Once you divide, you can solve the left limit, which equals -8. Now let's do the limit from the other side. As X approaches 2 from the right, X-2>0. Therefore you can rewrite the limit as... This limit, after you cancel, equals 8. Another success!! Can you discuss whether this limit exists? what about the one-sided limits?


 * You may be thinking that's awesome, but what do I do if there are some crazy symbols and such that I can't factor in the numerator? We've got it covered. **

Well, say you have a situation like this : Indeterminate form?!?! OH NO! jk you just have to rationalize the numerator From there you can simplify, like so: And then you can just substitute in 0 for x!!! <span style="color: #0000ff; font-family: 'Lucida Sans Unicode','Lucida Grande',sans-serif; font-size: 247%;">Now that we're pros at rationalizing the numerator, let's take a stab at rationalizing the <span style="color: #800000; font-family: 'Lucida Sans Unicode','Lucida Grande',sans-serif; font-size: 419.9%;"> denominator!!!!! Same basic process really.

You can see that when you plug in zero, the dreaded indeterminate form pops up again. When this happens, go back to your Algebra roots and conjugate!!

After you distribute and simplify, you should come out with this.

So as you can see, the x's cancel out, allowing you to plug zero into the equation, giving a limit of



**Now, what about **



The above graph is (sin x)/x.

To solve this limit, L'Hôpital's rule can be used. The numerator and denominator can be derived, to get: sin (x)/ x à cos (x)/x This can be written as: lim x à 0cos (x)/1 Here, the ** Direct Substitution ** method can be used to get: cos (0)/ 1 = 1/1 = 1 Therefore lim x à 0 sin (x)/x is 1.

**
 * Another way to look at this:



YOU DID IT!!

http://www.analyzemath.com/calculus/limits/find_limits_functions.html http://www.ehow.com/how_4962468_function-square-roots-expressions-numerator.html [] [|http://www.youtube.com/watch?v=Ve99biD1KtA]
 * SOURCES:**