Solving Limits: Uncovering the Answers

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In summary, the first limit problem does not exist because the behavior of (x+1)/(x-2) does not approach any value at x=2. The second limit problem can be solved by simplifying the numerator and then using the division rule, resulting in a limit of -1/4. The mistake in the first problem was forgetting to include the square in the denominator.
  • #1
Iron_Brute
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Two basic limit problems just solve the limit if it exists. I know the answers but I don't understand why the answers are what they are.
1.) lim (x^2-x-2) / (x-2)^2
x->2
2.) lim ((1/x)-(1/2)) / (x-2)
x->2

Answers were:
1.) DNE
2.) -1/4

I kind of understand that the first problem the denominator would be 0, but I thought if you factored the top and bottom you could find the limit of (x+1).

The second one I am completely stuck the only thing I could think to do was use the division rule but I would still have 0 for the denominator. If anyone could explain this it would be a big help.
 
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  • #2
1. Simplifying this expression by factoring the numerator is the correct approach. However, the behavior of (x+1)/(x-2) does not approach any value at x=2. To see this, try values of x slightly less than 2 and values of x slightly greater than 2. The graph is pretty much the graph of y = 1/x shifted 2 units to the right on the x-axis and 1 unit up on the y-axis. You can then add a rigorous epsilon-delta proof that the limit does not exist if necessary

2. Hint: Simplify the numerator by adding the two fractions.
 
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  • #3
I can not believe I did not think of adding the fractions together.

And I found out why I couldn't get the answer for the first problem. On my sheet I forgot to write the square on the denominator. And I wasn't thinking when I typed the problem. When I saw that you wrote (x+1)/(x-2) I went back to my sheet and saw that I did in fact forget the square.

Thank you so much for the help. Kind of ticked I did not see my mistake earlier though, spent a lot of time on the first problem trying to figure it out.
 

Related to Solving Limits: Uncovering the Answers

What is a limit in mathematics?

A limit in mathematics is a value that a function approaches as the input approaches a certain value. It is a fundamental concept in calculus that helps us understand the behavior of a function at a specific point.

Why is it important to solve limits?

Solving limits helps us determine the behavior of a function and its values at specific points. This is crucial in many real-world applications, such as optimization problems, physics, and engineering.

What are the different methods for solving limits?

There are several methods for solving limits, including direct substitution, factoring, using special limits, and using L'Hôpital's rule. The method used depends on the type of limit and the function involved.

How do you know if a limit does not exist?

If a function has different values approaching from the left and the right of a specific point, or if the function approaches infinity or negative infinity at a point, the limit does not exist.

Can limits be solved with just algebra?

In some cases, limits can be solved with just algebra by simplifying the function and plugging in the value of the variable. However, for more complex functions, other methods such as L'Hôpital's rule may be necessary.

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