Calculating limits by subbing in values

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In summary, the conversation discusses the concept of solving limits using substitution and how it may defeat the purpose of finding the limit. It also mentions the idea of continuity and the direct substitution property. The conversation ends with the clarification that the question has been answered by the professor in a lecture.
  • #1
tahayassen
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Doesn't solving limits by substituting values defeat the point? For example: you can solve a limit of a quadratic by just subbing in a x value. But how do we know limit as x approaches a is the same as f(a)?
 
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  • #2
Please clarify with an example.
 
  • #3
I'll try to take a stab at what you're getting at :)

For limits, we are concerned what y-value the function is headed for. It doesn't matter if there is a hole or something at that point, because the limit never actually equals the value x is headed towards. It can get infinitely close, however.

For some limits, we don't have to do much work or simplification. So, if you can plug in the x-value without the denominator being equal to 0, go for it!

Also, if lim x-> a of f(x) exits and equals f(a) , we call the function continuous (Assuming f(a) exists as well)
 
  • #4
Never mind.

My prof. answered my question in lecture today. It's called the direct substitution property.

http://web.viu.ca/wattsv/math121/Overheads/section14/directsubstitution.pdf [Broken]
 
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1. What is the purpose of calculating limits by subbing in values?

The purpose of calculating limits by subbing in values is to determine the behavior of a function at a specific point, also known as the limit point. This allows us to understand how a function approaches a certain value and can help us make predictions about the function's behavior.

2. How do you calculate limits by subbing in values?

To calculate limits by subbing in values, we simply plug in values that are close to the limit point on both sides of the function. We then observe the resulting outputs and determine if they approach a specific value or if they approach infinity. This can be done by hand or with the help of a graphing calculator.

3. What are the limitations of calculating limits by subbing in values?

Calculating limits by subbing in values may not always give us an accurate answer, especially when dealing with more complex functions. This method only gives us an approximation of the limit and may not work for functions that have discontinuities or vertical asymptotes.

4. Can we always use this method to calculate limits?

No, we cannot always use this method to calculate limits. It is only applicable when dealing with functions that are continuous at the limit point. If a function has a jump or a hole at the limit point, we cannot use this method to calculate the limit.

5. How can we check if the limit calculated by subbing in values is correct?

To check if the limit calculated by subbing in values is correct, we can use other methods such as the algebraic method, the graphing method, or the squeeze theorem. These methods can help us verify the limit and ensure that our calculation is accurate.

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