Solving a Limit Problem

In summary: In this case, the restriction is that a*b must be a positive number.So, in summary, the limit as n approaches ##\infty## is not true, and you cannot take the log of both sides of an equation to get a log of a quotient, nor can you take the log of a sum and get the sum of the logs.
  • #1
22990atinesh
143
1
Suppose there is a limit
##\lim_{n \to \infty} \frac{n^{1.74}}{n \times (\log n)^9}##
Taking logs both on numerator and denominator
##=\lim_{n \to \infty} \frac{1.74 \times \log n}{\log n + 9 \log \log n}##
What can we say about the limit as n approaches ##\infty##
 
Physics news on Phys.org
  • #2
How can you take log on both numerator and denominator. You are changing the question.
1/2= 0.5 is not the same as
log(1)/log(2) , which is 0.
 
  • #3
Raghav Gupta said:
How can you take log on both numerator and denominator. You are changing the question.
1/2= 0.5 is not the same as
log(1)/log(2) , which is 0.
In order to compare growth rate of two functions we can do that.
 
  • #4
Can you give an example?
Algebraically , in question you have stated how one can do that on numbers?
Is that a university( college ) level problem
Or a high school one?
 
  • #5
22990atinesh said:
Suppose there is a limit
##\lim_{n \to \infty} \frac{n^{1.74}}{n \times (\log n)^9}##
Taking logs both on numerator and denominator
##=\lim_{n \to \infty} \frac{1.74 \times \log n}{\log n + 9 \log \log n}##
What can we say about the limit as n approaches ##\infty##

22990atinesh said:
In order to compare growth rate of two functions we can do that.
In your first post (quoted above) you are claiming that ##\lim_{n \to \infty} \frac{n^{1.74}}{n \times (\log n)^9} = \lim_{n \to \infty} \frac{1.74 \times \log n}{\log n + 9 \log \log n}##. Most emphatically, this is NOT TRUE! You can take the log of both sides of an equation, but you cannot take the log of a quotient to get a quotient of logs, nor can you take the log of a sum and get the sum of the logs.

What IS true is that log(a * b) = log(a) + log(b), and that log(a/b) = log(a) - log(b), assuming suitable restrictions on the values of a and b.
 

1. What is a limit problem?

A limit problem is a mathematical concept that deals with the behavior of a function as its input approaches a particular value. It is used to determine the value that a function approaches as its input gets closer and closer to a specified value.

2. How do you solve a limit problem?

To solve a limit problem, you need to apply the rules and techniques of limits, such as algebraic manipulation, factoring, and substitution. You may also need to use special limit theorems, such as the Squeeze Theorem or the L'Hôpital's Rule, depending on the complexity of the problem.

3. What is the purpose of solving a limit problem?

The purpose of solving a limit problem is to understand the behavior of a function and to determine its value at a particular point. It is also useful in finding derivatives and evaluating integrals, which have many real-world applications in fields such as physics, engineering, and economics.

4. What are the common types of limit problems?

The common types of limit problems include finding the limit of a polynomial function, a rational function, a trigonometric function, an exponential function, or a logarithmic function. There are also more advanced limit problems involving indeterminate forms, such as 0/0 or ∞/∞, which require the use of limit theorems.

5. How can I practice and improve my skills in solving limit problems?

The best way to practice and improve your skills in solving limit problems is by working on a variety of problems, starting with simpler ones and gradually progressing to more challenging ones. You can find practice problems in textbooks, online resources, or by asking your teacher or peers for suggestions. It is also helpful to review the concepts and techniques of limits regularly and seek help when needed.

Similar threads

Replies
3
Views
1K
Replies
2
Views
656
Replies
6
Views
477
Replies
9
Views
1K
Replies
16
Views
2K
  • Calculus
Replies
3
Views
1K
Replies
15
Views
2K
Replies
24
Views
2K
  • Calculus
Replies
2
Views
1K
Back
Top