Limits and Rational Functions: What Rule Must Be Followed When Evaluating at 0?

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In summary, when evaluating a rational function at 0, it is important to define the function at that point to ensure consistency. In the case of the given problem, the "Input" and "Alternate Form" are not equal because the "Input" is not defined at s=0, while the "Alternate Form" is defined. However, the limit of the "Input" as s goes to 0 does exist and is equal to the "Alternate Form". Therefore, in the limit, the two forms are equal, but as rational forms, they are not. This highlights the importance of properly defining a function when evaluating it at a specific point.
  • #1
Ry122
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what rule are you supposed to follow when you evaluate a rational function at 0? eg in this problem if you evaluate at s=0 for the one under "result" it will be different from the value obtained for the one under "alternate forms" http://www.wolframalpha.com/input/?i=(B/(m*(s^2)+k))/(1+(r*s^2+t*s)/(k*s+m*s^3))

What do I have to do to keep things consistent?
 
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  • #2
Ry122 said:
what rule are you supposed to follow when you evaluate a rational function at 0? eg in this problem if you evaluate at s=0 for the one under "result" it will be different from the value obtained for the one under "alternate forms" http://www.wolframalpha.com/input/?i=(B/(m*(s^2)+k))/(1+(r*s^2+t*s)/(k*s+m*s^3))

What do I have to do to keep things consistent?

What exactly do you mean by "evaluation"?
At s=0 your initial input involves division by zero...
Well if you calculate a limit then everything is consistent.
 
  • #3
This is the danger of programs like wolfram alpha. They're very good, but you got to be able to interpret the solution.

In this case, the "Input" and the "Alternate Form" are not equal. Indeed, the problem is that you can define by 0. In the "Input", if you put s=0, then you define by 0, so the "Input" is not defined at s=0.
But the "Alternate form" is defined at s=0 (whenever k+t is nonzero).

So the two forms are not equal since one is defined at s=0 and the other is not.

However, the limit of the "Input" as s goes to 0 does exist and does equal the "Alternate Form". So in the limit, the expressions are equal. But as rational forms, the expressions are not equal.
 

1. What is the definition of a limit?

A limit is a mathematical concept that represents the value that a function or sequence approaches as its input or index approaches a certain value.

2. How do you evaluate a limit?

To evaluate a limit, you can use various techniques such as direct substitution, factoring, rationalization, and L'Hopital's rule. Additionally, you can graph the function or use a table of values to estimate the limit.

3. What is the difference between a one-sided limit and a two-sided limit?

A one-sided limit only considers the values approaching from one side of the input value, while a two-sided limit considers the values approaching from both sides of the input value. One-sided limits are denoted by a plus or minus sign in the limit notation, while two-sided limits are denoted by just the limit notation.

4. Can a limit exist even if the function is not defined at that point?

Yes, a limit can exist even if the function is not defined at that point. This is known as a removable discontinuity, where the function has a hole at that point but can still approach a certain value as the input approaches that point.

5. What are the common types of limits encountered in calculus?

The common types of limits encountered in calculus include limits at infinity, limits involving trigonometric functions, and limits involving exponential and logarithmic functions. Additionally, there are also limits involving indeterminate forms such as 0/0 and infinity/infinity.

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