Rewriting Limit Problem with Conjugate - Explained | Get a Better Understanding

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The book suggests rewriting it using the conjugate because it can help simplify the expression and make it easier to evaluate the limit. In summary, the book recommends rewriting the problem using the conjugate to simplify the expression and make it easier to evaluate the limit.
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CrossFit415
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x[itex]\stackrel{Lim}{\rightarrow}[/itex]infinity ([itex]\sqrt{x^{2}+1}[/itex])[itex]-1[/itex]

Couldn't I just inset 0 for x and end up getting [itex]\sqrt{1}[/itex]?

The book tells me to rewrite this problem by its conjugate.

My question is why? Thanks
 
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CrossFit415 said:
x[itex]\stackrel{Lim}{\rightarrow}[/itex]infinity ([itex]\sqrt{x^{2}+1}[/itex][itex]-1[/itex]

Couldn't I just inset 0 for x and end up getting [itex]\sqrt{1}[/itex]?

The book tells me to rewrite this problem by the conjugate.

My question is why? Thanks
As you have written it,
[tex]\lim_{x \to \infty}\sqrt{x^2 + 1} - 1 = \infty[/tex]
You seem to be a little confused in this problem. Why would you put in zero for x when the limit is as x goes to infinity?
 

FAQ: Rewriting Limit Problem with Conjugate - Explained | Get a Better Understanding

What is a limit problem?

A limit problem is a mathematical concept that involves finding the value that a function approaches as the input approaches a certain value. It is used to describe the behavior of a function at a specific point or as the input approaches a specific value.

What is a conjugate in a limit problem?

In a limit problem, a conjugate is the expression that results from changing the sign between two terms. It is used in solving limit problems involving fractions, as it helps to simplify the expression and make it easier to evaluate the limit.

How do you rewrite a limit problem with conjugates?

To rewrite a limit problem with conjugates, you need to find the conjugate of the original expression by changing the sign between the two terms. Then, use the conjugate to rationalize the original expression by multiplying the top and bottom of the fraction by the conjugate. This will result in a new expression that does not have any radicals or complex numbers.

Why is rewriting limit problems with conjugates useful?

Rewriting limit problems with conjugates is useful because it allows you to simplify the expression and make it easier to evaluate the limit. It also helps to eliminate any radicals or complex numbers, which can make the expression more complicated to solve.

Are there any limitations to using conjugates in limit problems?

While using conjugates can be a helpful strategy in solving limit problems, it is not always necessary or the most efficient method. In some cases, there may be other approaches that are easier or more straightforward. It is important to understand the concept of conjugates and when it is appropriate to use them in solving limit problems.

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