LimitsAnalytically1

**Finding Limits Analytically!** **Created by: Cristina Iraira, Harmony Ward, Norman Cyr** **First of all, what exactly is limit?** **A limit is a certain value to which a function approaches. Finding a limit means finding what Y is as X approaches a certain number. Finding a limit can be done by a few different ways. You can plug and chug or you can simplify. Here are some examples that might help you understand what a limit is. As the polygon inscribed in the circle is getting more and more sides, the polygon is becoming the circle it is inscribed in, but it will never get there. There are a few ways to find limits like plugging and chugging and simplifying!**







**[|Cool Math Lessons-Calculus]**

** Plugging and Chugging Plugging and chugging is simple. Basically you would be taking that X as it approaches infinitely and substituting that infinitely symbol in for X in the function. To see the table, look at the link below the problem. Here is an example: ****Lim F(X) X^3+X^2+X+9 X->2 ** **Plug in that 2 for X and chug away!** **Lim F(X) 2^3+2^2+2+9=23**

** Here is an example: ** **Lim F(X) X^2+9 X->5 ** **Plug in that 5 for X and chug away!** **Lim F(X) 5^2+9=34**

**Factoring** **Factoring is a little bit harder, but not impossible to understand. Basically you would factor the denominator and then cancel out the common factors from both the numerator and the denominator.**

**Here is an example by simplifying:**

**F(X)= X-2 / X^2-4 X->2 **  **F(X)= X-2 / (X-2) (X+2)** **F(X)= 1 / (X+2)** **F(X)= 1 / (2+2)** **F(X)=1/4**

**F(X)= X-2 / X^2-4=1/4As F(X) approaches 2, X equals 1/4. To find the 1/4, you have to start out by factoring the denominator of the function. So, you will end up with X-2 / (X-2) (X+2). You will then cancel out the (X-2)'s in both the numerator and the denominator. After this is done, you will be left with 1/(X+2). So, remember as that F(X) approaches 2? You have to plug in that 2 for X, so your function will look like 1/(2+2). The numerator is still 1 and the sum of the denominator equals 4. 1/4 would be the answer of F(X)=X-2/ X^2-4 as X approaches 2.**

** Here is another harder example: ** **Lim F(X)= (X^2-X-6) / (X^2+6X+8)** **X->-2** ** lim x → − 2 ( X^ 2 − X − 6 ) / (X^2+6X+8)  First, try to plug in − <span class="mn" style="font-family: MathJax_Main; font-size: 20px;">2  for <span class="mi" style="font-family: MathJax_Math; font-size: 20px;">x. <span class="mo" style="font-family: MathJax_Main; font-size: 20px;">((− <span class="mn" style="font-family: MathJax_Main; font-size: 20px;">2 <span class="mo" style="font-family: MathJax_Main; font-size: 20px;">) <span class="mn" style="font-family: MathJax_Main; font-size: 14px;">^2 <span class="mo" style="font-family: MathJax_Main; font-size: 20px;">−(− <span class="mn" style="font-family: MathJax_Main; font-size: 20px;">2 <span class="mo" style="font-family: MathJax_Main; font-size: 20px;">)− <span class="mn" style="font-family: MathJax_Main; font-size: 20px;">6 <span class="mo" style="font-family: MathJax_Main; font-size: 20px;">) / ((− <span class="mn" style="font-family: MathJax_Main; font-size: 20px;">2 <span class="mo" style="font-family: MathJax_Main; font-size: 20px;">)^ <span class="mn" style="font-family: MathJax_Main; font-size: 14px;">2 <span class="mo" style="font-family: MathJax_Main; font-size: 20px;">+ <span class="mn" style="font-family: MathJax_Main; font-size: 20px;">6 <span class="mo" style="font-family: MathJax_Main; font-size: 20px;">(− <span class="mn" style="font-family: MathJax_Main; font-size: 20px;">2 <span class="mo" style="font-family: MathJax_Main; font-size: 20px;">)+ <span class="mn" style="font-family: MathJax_Main; font-size: 20px;">8 <span class="mo" style="font-family: MathJax_Main; font-size: 20px;">)= <span class="mn" style="font-family: MathJax_Main; font-size: 20px;">0 <span class="mo" style="font-family: MathJax_Main; font-size: 20px;">/ <span class="mn" style="font-family: MathJax_Main; font-size: 20px;">0. Since it equals <span class="mn" style="font-family: MathJax_Main; font-size: 20px;">0 <span class="mo" style="font-family: MathJax_Main; font-size: 20px;">/ <span class="mn" style="font-family: MathJax_Main; font-size: 20px;">0, we must use another method to solve the limit. Factor the numerator and the denominator and find a factor that is common to the numerator and the denominator of the limit. <span class="mo" style="font-family: MathJax_Main; font-size: 20px;">lim <span class="mi" style="font-family: MathJax_Math; font-size: 14px;">x <span class="mo" style="font-family: MathJax_Main; font-size: 14px;">→ − <span class="mn" style="font-family: MathJax_Main; font-size: 14px;">2 <span class="mo" style="font-family: MathJax_Main; font-size: 20px;">(( <span class="mi" style="font-family: MathJax_Math; font-size: 20px;">x <span class="mo" style="font-family: MathJax_Main; font-size: 20px;">− <span class="mn" style="font-family: MathJax_Main; font-size: 20px;">3 <span class="mo" style="font-family: MathJax_Main; font-size: 20px;">) ( <span class="mi" style="font-family: MathJax_Math; font-size: 20px;">x <span class="mo" style="font-family: MathJax_Main; font-size: 20px;">+ <span class="mn" style="font-family: MathJax_Main; font-size: 20px;">2  <span class="mo" style="font-family: MathJax_Main; font-size: 20px;"> ) ) / (( <span class="mi" style="font-family: MathJax_Math; font-size: 20px;">x <span class="mo" style="font-family: MathJax_Main; font-size: 20px;">+ <span class="mn" style="font-family: MathJax_Main; font-size: 20px;">4 <span class="mo" style="font-family: MathJax_Main; font-size: 20px;">) (  <span class="mi" style="font-family: MathJax_Math; font-size: 20px;">x <span class="mo" style="font-family: MathJax_Main; font-size: 20px;">+ <span class="mn" style="font-family: MathJax_Main; font-size: 20px;">2  <span class="mo" style="font-family: MathJax_Main; font-size: 20px;"> ) )  Now, we can cancel away the <span class="mo" style="font-family: MathJax_Main; font-size: 20px;">( <span class="mi" style="font-family: MathJax_Math; font-size: 20px;">x <span class="mo" style="font-family: MathJax_Main; font-size: 20px;">+ <span class="mn" style="font-family: MathJax_Main; font-size: 20px;">2 <span class="mo" style="font-family: MathJax_Main; font-size: 20px;">)  in the numerator and denominator.** ** <span class="mo" style="font-family: MathJax_Main; font-size: 20px;">lim <span class="mi" style="font-family: MathJax_Math; font-size: 14px;">x <span class="mo" style="font-family: MathJax_Main; font-size: 14px;">→ − <span class="mn" style="font-family: MathJax_Main; font-size: 14px;">2 <span class="mo" style="font-family: MathJax_Main; font-size: 20px;">( <span class="mi" style="font-family: MathJax_Math; font-size: 20px;">x <span class="mo" style="font-family: MathJax_Main; font-size: 20px;">− <span class="mn" style="font-family: MathJax_Main; font-size: 20px;">3 <span class="mo" style="font-family: MathJax_Main; font-size: 20px;">) / ( <span class="mi" style="font-family: MathJax_Math; font-size: 20px;">x <span class="mo" style="font-family: MathJax_Main; font-size: 20px;">+ <span class="mn" style="font-family: MathJax_Main; font-size: 20px;">4 <span class="mo" style="font-family: MathJax_Main; font-size: 20px;">) Now, we will try to plug in the -2 and chug away.<span class="mo" style="font-family: MathJax_Main; font-size: 20px;">(− <span class="mn" style="font-family: MathJax_Main; font-size: 20px;">2 <span class="mo" style="font-family: MathJax_Main; font-size: 20px;">− <span class="mn" style="font-family: MathJax_Main; font-size: 20px;">3 <span class="mo" style="font-family: MathJax_Main; font-size: 20px;">) / (− <span class="mn" style="font-family: MathJax_Main; font-size: 20px;">2 <span class="mo" style="font-family: MathJax_Main; font-size: 20px;">+ <span class="mn" style="font-family: MathJax_Main; font-size: 20px;">4 <span class="mo" style="font-family: MathJax_Main; font-size: 20px;">)=− <span class="mn" style="font-family: MathJax_Main; font-size: 20px;">5 <span class="mo" style="font-family: MathJax_Main; font-size: 20px;">/ <span class="mn" style="font-family: MathJax_Main; font-size: 20px;">2 **



** Here are some videos that might also help you learn how to find a limit analytically: **

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**Limits that do not exist** **A limit cannot exist at an X value if the Y values approach different numbers from each side.** **To find the coordinates, take the number that X approaches and that is your X value. To find the Y value, take that number that X approaches and put it in for X and solve.** ** A example to help you understand: **

<span style="background-color: transparent; display: block; text-align: center; text-decoration: none; vertical-align: 0px;">**<span style="background-color: #ffffff; font-family: MathJax_Math; vertical-align: 0px;">F(X)={ X-1 if X< 2 (2,1) -open ** **<span class="mi" style="background-color: transparent; font-family: MathJax_Math; text-decoration: none; vertical-align: 0px;"> { (X-2)^2 +3 if X ≥ 2 (2,3) -closed**



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