Rules of limits (breaking a larger limit into 2 smaller ones)

In summary, the algebraic limit theorem states that the sum of two limits, if they exist and the sum is well-defined, is the limit of the sum. This is because addition is a continuous function. One-sided limits are important in determining the existence of a two-sided limit, as both one-sided limits must exist and be equal for the two-sided limit to exist. Therefore, splitting (a + b)/h into a/h + b/h may not be helpful if the limits of the individual terms do not exist.
  • #1
Moogie
168
1
Hi

What rule or theorem of limits says that you can do this with a limit

lim(h->0) (a+b)/h =
lim(h->0) a/h + lim(h->0) b/h

The book i am reading has just done something like that to a limit without saying why you can do that

thanks
 
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  • #2
Its called the algebraic limit theorem
 
  • #3
This is just the limit law for +: the sum of two limits, if they exist and the sum is well-defined, is the limit of the sum.


More generally, it's just that + is a continuous function.
 
  • #4
If the value of a + b isn't zero, the limit does not exist.

Have you come across the concept of one-sided limits yet? For example,
[tex]\lim_{h \to 0^+} 1/h = \infty[/tex]
and
[tex]\lim_{h \to 0^-} 1/h = -\infty[/tex]

In the first limit, h approaches zero from the positive side. In the second limit, h approaches zero from the negative side.

An important concept of limits is that in order for the two-sided limit to exist, both one-sided limits must exist and must be equal. In the example I gave, the two one-sided limits are different, so
[tex]\lim_{h \to 0} 1/h \text{does not exist}[/tex]

My point is that it might not help to split (a + b)/h into a/h + b/h if the limits of the terms on the right don't exist.
 

1. What are the rules for breaking a larger limit into two smaller ones?

The rules for breaking a larger limit into two smaller ones are as follows:

  1. If the limit of f(x) as x approaches a is equal to L, and the limit of g(x) as x approaches a is equal to M, then the limit of f(x) + g(x) as x approaches a is equal to L + M.
  2. Similarly, if the limit of f(x) as x approaches a is equal to L, and the limit of g(x) as x approaches a is equal to M, then the limit of f(x) - g(x) as x approaches a is equal to L - M.
  3. If the limit of f(x) as x approaches a is equal to L and the limit of g(x) as x approaches a is equal to M, then the limit of f(x) * g(x) as x approaches a is equal to L * M.
  4. If the limit of f(x) as x approaches a is equal to L, and the limit of g(x) as x approaches a is equal to M (with M not equal to 0), then the limit of f(x) / g(x) as x approaches a is equal to L / M.
  5. Lastly, if the limit of f(x) as x approaches a is equal to L, and the limit of g(x) as x approaches a is equal to M, then the limit of f(x)^n as x approaches a is equal to L^n.

2. How do these rules apply to breaking a larger limit into two smaller ones?

These rules allow us to break down a complicated limit into smaller, simpler limits that are easier to solve. By applying these rules, we can manipulate the smaller limits and then combine them to find the overall limit of the larger expression.

3. Can these rules be applied to all limits?

Yes, these rules can be applied to any limit, as long as the individual limits involved in the expression exist and are finite. If the individual limits do not exist or are infinite, then these rules cannot be applied.

4. Are there any exceptions to these rules?

Yes, there are some exceptions to these rules. One exception is when we are dealing with limits involving trigonometric functions. In these cases, we may need to use other techniques, such as trigonometric identities, to break down the limit into smaller ones. Another exception is when we are dealing with limits involving absolute values, where we may need to consider the possibility of the limit approaching from the left and the right separately.

5. How can these rules be helpful in limit calculations?

These rules can be extremely helpful in simplifying complex limit expressions and making them easier to solve. By breaking a larger limit into smaller ones, we are able to apply basic arithmetic operations and known limit values to find the overall limit. This can save time and effort in limit calculations and also provide a better understanding of the limit concept.

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