# Trouble with the following limit

• devious_
In summary, a limit in mathematics is the value that a function approaches as its input approaches a specific value or approaches infinity. When a limit is undefined, it means the function does not approach a single value or approaches different values from the left and right sides of the input. To evaluate a limit algebraically, techniques such as direct substitution, factoring, rationalization, and the use of trigonometric identities can be used. A one-sided limit only considers the behavior of the function from one side of the input, while a two-sided limit takes into account the behavior from both sides. A limit can also be found graphically by observing the behavior of the function near the input value.
devious_
I'm having trouble with the following limit:

$$\lim_{x \rightarrow 0} \frac{1 - \sqrt{1 - 4x^2}}{x^2}$$

At the back of the book (Apostol Volume I), it says the answer is 1/2, but I get 2. Can anyone clarify?

Thanks.

multiply by 1 :
$$\frac{1 + \sqrt{1-4x^2}}{1 + \sqrt{1-4x^2}}$$

Seems to me you are correct

Yes,even l'Hôpital approves you guys.A typo in the book,nothing sensational.

Daniel.

Good to know. Thanks.

## 1. What is a limit in mathematics?

A limit is a fundamental concept in mathematics that represents the value that a function approaches as its input approaches a specific value or approaches infinity.

## 2. What does it mean when a limit is undefined?

A limit is undefined if the function does not approach a single value or approaches different values from the left and right sides of the input.

## 3. How do you evaluate a limit algebraically?

To evaluate a limit algebraically, you can use various techniques such as direct substitution, factoring, rationalization, and the use of trigonometric identities.

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

A one-sided limit only considers the behavior of the function from one side of the input, either the left or the right. A two-sided limit takes into account the behavior of the function from both the left and right sides of the input.

## 5. Can a limit be found graphically?

Yes, a limit can be found graphically by observing the behavior of the function near the input value. The value that the function approaches as the input approaches the specific value is the limit.

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