Solving a Calculus Problem: Charting the Limits

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In summary, the speaker is struggling to understand calculus and has asked for help. They are working on a problem involving limits and evaluating a function at specific points. The book has provided a chart with these evaluations, but the speaker is confused about how the book got these numbers. They mention using factoring and substitution to solve limits, but are also curious about L'hopital's rule, which is a more advanced technique for solving limits. Another speaker clarifies that factoring can work in this case and explains the concept of L'hopital's rule. The speaker expresses their dislike for using charts and hopes to just use substitution and factoring in the future.
  • #1
Johnnycab
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Hi everyone I am COmpletly oblivious when it comes to calculus and I am going to try and learn it myself. i took a calculus course in college and got a C, i remember some things but i pretty much cheated. That was a couple of years ago, and well now i want to learn it :confused: . Anyways here is my problem

This is an example, not my own work

lim (x^2 - 4) / (5x - 10)
x->2

when i sub 2 into x i get 0/0, so I am asked to make a chart I am given

x **********(x^2 - 4) / (5x - 10) right side limit
3 ****1.0
2.5 ****** 0.9
2.1 ****** 0.82
2.01 ******* 0.802
2.001 ********* 0.8002x ********(x^2 - 4) / (5x - 10) left side limit
1 ******* 0.6
1.5 ******* 0.7
1.9 ****** 0.78
1.99 ***** 0.798
1.999 ****** 0.7998

what i don't understand is how the book got these numbers, like on the right side 3 appears to equal 1 and on the left side 1 appears to equal (.6). I am saying appears to equal because i didnt see a equal sign. what i think i understand is that on the left side its showing x getting closer to a. But i don't know

I hope this is enough information on my part to count, i would really appreciate someone to help me understand

thank you in advance

PS- the astriks are there to make space between the numbers
 
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  • #2
They are showing you x values around 2, the value that you really care about, and then showing you what the function evaluates to for those values of x.
 
  • #3
that sounds right i guess, the answer ends up being 0.8, and not 0/0. How the auther of the example got that i won't know.

Could i just factor the equation instead?

thank you for your feedback btw :)
 
  • #4
All that the book has done is evaluate the given function at the specified points. Try it. It is not clear to me why you have trouble with f(1)=.6 and f(3)=1, separate points, separate results. The best way to see this is to plug in the numbers for yourself.
 
  • #5
0/0 is undefined in our mathematics. In the event of an undefined solution you need use a different technique. You can either try to factor out stuff or in this case you have to use what is called l'hopital's rule (I am sure your text will go into that in the future). What the author was doing was showing you that as your input values (x) gets closer and closer to 2, the solution gets closer and closer to .8, which will eventually lead to the conclusion that the solution (the limit) is in fact .8.

With limits your not asking what is the value of the equation when you plug in this input (2, in our example), but rather what is it approaching when you approach the input (2).

This can be seen easily by asking the following:

lim (x+1)(x-3)/(x-3)
x->3

If you were to graph this it would look like (x+1) (because the (x-3)/(x-3) terms would factor out) but since there is the (x-3)/(x-3) the solution is undefined (0/0) at x=3 and therefore there is a hole in the line at that value (there is an undefined solution at x=3). But you can see that the equation (x+1) approaches => (3+1) at x = 3 so you could say the limit is 4 at x=3.

Does that make sense (except for the l'hopital's rule that is a little advanced at this stage.)
 
  • #6
D_Dean said:
You can either try to factor out stuff or in this case you have to use what is called l'hopital's rule)

No, factoring works in this case. 5x-10, and x2-4 share a common factor of x-2. So you don't HAVE to use L'hopital's rule.
 
  • #7
-Cool thank you everyone for your input, i very much appreciate it
-so when I am doing limits i can solve equations by
1. substituting for x,
2. factoring
3. using a chart
-i really don't like #3 - charts i hope in the future i can just substitute and factor :cool:
-what is L'hopital's rule?
 
  • #8
Johnnycab said:
-what is L'hopital's rule?

If you're working through a calculus book it should be presented later in the book, probably as an application of differentiation, I don't think that I can expplain it very well, however, so I will give you the link to the wikipedia page on this rule, and someone else may also be able to explain it.

http://en.wikipedia.org/wiki/L'Hôpital's_rule
 

FAQ: Solving a Calculus Problem: Charting the Limits

What is a limit in calculus?

A limit in calculus is a fundamental concept that represents the value that a function or sequence approaches as its input or index approaches a certain value. It is used to describe the behavior of a function near a specific point.

How do you solve a calculus problem involving limits?

To solve a calculus problem involving limits, you need to first determine the type of limit (left-sided, right-sided, or two-sided) and then apply the appropriate limit laws and rules. You may also need to use techniques such as L'Hopital's rule, squeeze theorem, or the definition of a limit to evaluate the limit.

Why is it important to understand limits in calculus?

Understanding limits in calculus is crucial because it is the foundation for many other concepts in calculus, such as continuity, derivatives, and integrals. It also helps us understand the behavior of a function and make predictions about its values at certain points.

Can you provide an example of solving a calculus problem involving limits?

Sure, let's say we have the function f(x) = (x^2 - 4)/(x - 2). To find the limit as x approaches 2, we can factor the numerator and simplify to get f(x) = x + 2. Then, we can simply plug in x = 2 to get the limit as 4.

Are there any common mistakes to avoid when solving calculus problems involving limits?

Yes, some common mistakes to avoid when solving calculus problems involving limits include forgetting to check for the existence of a limit, applying the wrong limit laws, and not simplifying the expression before finding the limit. It is also important to be careful when dealing with indeterminate forms such as 0/0 or ∞/∞, as they require special techniques to evaluate.

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