# Derivative using the limit definition (without using L'Hospital's rule)

• MHB
• goody1
In summary, the person is asking for help finding a derivative without using L'Hospital's rule. They have used the definition but are unsure of what to do next. Someone suggests using a series for the arctangent function and the person realizes that the limit of arctan(3pi/x) does not exist, therefore the derivative also does not exist. The person thanks Klaas van Aarsen and realizes that the limit of arctan(x) does exist, which can be used to find the derivative.
goody1
Hello everybody, could you help me with this problem please? I have to find a derivative in x0 of this function (without using L'Hospital's rule):
View attachment 9694

I used the definition View attachment 9695, but I don't know what to do next. Thank you.

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goody said:
Hello everybody, could you help me with this problem please? I have to find a derivative in x0 of this function (without using L'Hospital's rule):I used the definition , but I don't know what to do next. Thank you.

First of all, the denominator is x - x_0, not x - 0.

If you can't use L'Hospital's Rule (which, by the way, is a pointless constraint designed to make life more difficult for the person doing the work), then I'd advise using a series for the arctangent function.

Note that $f(x_0)=f(0)$ has been defined as $0$.

So we have:
$$f'(x_0)=\lim_{x\to x_0}\frac{\pi x^2+x\arctan\frac{3\pi}x-f(x_0)}{x-x_0} =\lim_{x\to 0}\frac{\pi x^2+x\arctan\frac{3\pi}x-f(0)}{x-0} =\lim_{x\to 0}\Big(\pi x+\arctan\frac{3\pi}x\Big)$$
Furthermore:
$$\lim_{x\to 0^+}\arctan\frac{3\pi}x = \frac\pi 2$$
$$\lim_{x\to 0^-}\arctan\frac{3\pi}x = -\frac\pi 2$$
So $\lim\limits_{x\to 0}\arctan\frac{3\pi}x$ does not exist, and therefore $f'(x_0)$ does not exist either.

I know using L'Hospital's rule would be easy way to solve it but we haven't learned it yet so we're forced to find another ways.

Anyways, thank you so much Klaas van Aarsen, now when you showed me it looks so simple.

$$\displaystyle \lim_{\theta \to \pi/2} \tan(\theta)= \infty$$ so $$\displaystyle \lim_{x \to\infty} \arctan(x)=\pi/2$$. That limit certainly does exist!

## 1. What is the limit definition of a derivative?

The limit definition of a derivative is the mathematical formula that describes the rate of change of a function at a specific point. It is written as the limit of the difference quotient as the change in x approaches 0.

## 2. How do you find the derivative using the limit definition?

To find the derivative using the limit definition, you need to follow these steps:
1. Write out the difference quotient formula: (f(x+h) - f(x)) / h
2. Simplify the equation as much as possible
3. Take the limit as h approaches 0
4. The resulting value is the derivative at the given point.

## 3. Why do we use the limit definition to find derivatives?

The limit definition is used to find derivatives because it is the most fundamental and accurate way to determine the instantaneous rate of change of a function at a specific point. It also allows us to find the derivative of any type of function, not just polynomial or rational functions.

## 4. Can we use L'Hospital's rule to find derivatives instead of the limit definition?

Yes, L'Hospital's rule can be used to find derivatives, but it is not always the most efficient method. It is typically used when the limit definition is difficult to evaluate or when the function is in an indeterminate form.

## 5. What are the common mistakes when finding derivatives using the limit definition?

Some common mistakes when finding derivatives using the limit definition include:
- Forgetting to take the limit as h approaches 0
- Not simplifying the difference quotient enough
- Using incorrect algebraic manipulations
- Misinterpreting the notation and not substituting the correct values
To avoid these mistakes, it is important to carefully follow the steps and double check your work.

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