"Did I Solve the Limit Value Question on Today's Exam Correctly?

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In summary, the conversation was about a question that appeared on an exam and the use of the "sum to product formula" to find a limit value. The limit was found to be zero using the formula, which involved evaluating the limit of a cosine expression. The conversation also mentioned the boundedness of cosine and sine in relation to finding the limit.
  • #1
KLscilevothma
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This question appeared in today's exam but I think I did it wrongly. I use the "sum to product formula", but I can't find a limit value and say it doesn't exist. Am I correct?

[tex]\lim_{n\rightarrow\infty} cox \sqrt{2004 + x} - cos \sqrt{x} [/tex]
 
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  • #2
I take it that you are to find the limit when x->inf, not n->inf.
The limit is zero:
Note that sqrt(a+x)=sqrt(x)*sqrt(1+(a/x)) approx. sqrt(x)+1/2(a/sqrt(x)), when a<<x

Hence, we may write the original cosine as cos(sqrt(x)+e),
where e->0 as x->inf.

Using sum-to-product, we have to evaluate the limit of:
cos(sqrt(x))*(cos(e)-1)-sin(e)*sin(sqrt(x)).

Since cos(sqrt(x)), sin(sqrt(x)) are bounded by 1, we see that the whole expression goes to 0.
 
  • #3
I see. Thanks
 

What is a limit value?

A limit value, also known as a limit, is a fundamental concept in mathematics that represents the value that a function or sequence approaches as the input approaches a certain point. It is a way to describe the behavior of a function near a specific input without actually evaluating the function at that point.

How is a limit value calculated?

A limit value is typically calculated using the limit definition, which involves taking the limit of a function as the input approaches a specific point. This can be done algebraically, graphically, or numerically, depending on the function and the specific point being considered. In some cases, the limit value may not exist or may require more advanced mathematical techniques to calculate.

Why are limit values important?

Limit values are important because they allow us to study the behavior of functions and sequences in a precise and rigorous manner. They are also essential in many areas of mathematics, including calculus, differential equations, and real analysis. In addition, limit values have many practical applications in fields such as physics, engineering, and economics.

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

A one-sided limit is a limit that is calculated as the input approaches a specific point from only one direction, either the left or the right. A two-sided limit, on the other hand, is a limit that is calculated as the input approaches a specific point from both the left and the right. Two-sided limits are often used to determine if a limit value exists, while one-sided limits can help us understand the behavior of a function near a specific point.

How do limit values relate to continuity?

A function is said to be continuous at a point if the limit of the function at that point exists and is equal to the value of the function at that point. In other words, continuity is a special case of a limit value where the two values are equal. This means that limit values are closely related to the concept of continuity, and understanding limit values is essential for understanding continuity and related concepts such as differentiability.

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