Limit comparisons (reviewing my Calc 1)

In summary, the conversation discusses the comparison of two functions and their respective limits as one approaches from the right. The speaker explains that the first graph is a vertical stretch and horizontal shift of the second graph. They also mention that it is acceptable to compare limits of similar functions that are related by a change of variable.
  • #1
rocomath
1,755
1
So recently I been reviewing my graphing along with my Calculus 1.

I'm currently reading this problem and I was thinking, why not just compare it to a graph that I already know.

[tex]\lim_{x \rightarrow 5^{+}}\frac{6}{x-5}[/tex]

with comparison to whose limit also goes to Infinity as it approaches from the right.

[tex]\lim_{x \rightarrow 0^{+}}\frac{1}{x}[/tex]

Basically, my interpretation is that the first graph is simply stretched vertically by 6 and shifted 5 units from the graph of 1\x.

Is it ok for me to compare limits of similar functions such as the ones I just compared?
 
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  • #2
It's ok with me. Your description of the transformation from one to the other is correct. The sense in which these functions are 'similar' is that they are related by a change of variable.
 

Related to Limit comparisons (reviewing my Calc 1)

1. What is the purpose of using limit comparisons in calculus?

The purpose of using limit comparisons in calculus is to determine the behavior of a function as it approaches a certain value or limit. It allows us to compare the growth or decline rate of a function to that of a simpler function, making it easier to analyze and understand the behavior of more complex functions.

2. How do you perform a limit comparison test?

To perform a limit comparison test, you must first identify the function you want to analyze and the simpler function you want to compare it to. Then, take the limit of the quotient of the two functions as x approaches the limit in question. If the limit is equal to a finite nonzero number, then the two functions have the same behavior at that limit.

3. What is the difference between limit comparisons and direct comparisons?

The main difference between limit comparisons and direct comparisons is that limit comparisons focus on the behavior of a function as it approaches a specific limit, while direct comparisons look at the overall behavior of a function. Limit comparisons are often used to determine the convergence or divergence of a series, while direct comparisons are used to compare the growth or decline rates of two functions.

4. Can limit comparisons be used for all functions?

No, limit comparisons can only be used for functions that are positive and decreasing. If a function is negative or increasing, then a different method, such as direct comparisons or the ratio test, should be used to analyze its behavior.

5. How do limit comparisons help in finding the convergence or divergence of a series?

Limit comparisons can help in finding the convergence or divergence of a series by comparing it to a known convergent or divergent series. By taking the limit of the quotient of the two series, we can determine if the original series behaves similarly to the known series and thus determine its convergence or divergence.

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