# Solving a Limit Involving tanx and pi/4

• Glissando
Do you see how this is done?In summary, the problem is finding the limit of (tanx-1)/(x-pi/4) as x approaches pi/4. This can be rewritten using the formula tanx = sinx/cosx and simplified to [(sinx-cosx)/cosx]/(x-pi/4). From here, we can use the formula tan(x)=tan((x-\pi/4)+\pi/4) and apply this to simplify the expression further in terms of x-\pi/4. The goal is to write everything in function of x-\pi/4.
Glissando

## Homework Statement

Find the limit:

lim (tanx-1)/(x-pi/4)
x->pi/4

tanx = sinx/cosx

## The Attempt at a Solution

lim (sinx/cosx-1)/(x-pi/4)
x->pi/4

lim [(sinx-cosx)/cosx]/(x-pi/4)
x->pi/4

I have no idea what to do after this ): I also tried squaring the whole function and getting tan2x-1 = sec2x, but I get so lost on what to do with pi/4 because it just keeps becoming undefined!

Hi Glissando!

I take it you're not allowed to use L'hopitals rule?? In that case, I would first write

$$\tan(x)=\tan((x-\pi/4)+\pi/4)$$

and work that out. That way you can write everything in function of $x-\pi/4$.

micromass said:
Hi Glissando!

I take it you're not allowed to use L'hopitals rule?? In that case, I would first write

$$\tan(x)=\tan((x-\pi/4)+\pi/4)$$

and work that out. That way you can write everything in function of $x-\pi/4$.

Hi micromass,

Thanks for your quick response! I'm not too sure what you mean by working it out...do I plug that back into the original equation? Am I solving for x?

Thanks!

Well, you know formula's for $\tan(\alpha+\beta)$. So apply these formula's on

$$\tan((x-\pi/4)+\pi/4)$$

Our goal is to write everything in function of $x-\pi/4$

## 1. What is a limit involving tanx and pi/4?

A limit involving tanx and pi/4 is a mathematical concept that involves finding the value that a function approaches as the input (x) approaches a specific value (pi/4). In this case, the function is tangent (tanx) and the specific value is pi/4.

## 2. How do you solve a limit involving tanx and pi/4?

To solve a limit involving tanx and pi/4, you can use trigonometric identities and algebraic manipulation to rewrite the function in a form that allows you to directly substitute the specific value (pi/4) for x. You can then evaluate the limit by plugging in the value and simplifying the expression.

## 3. What is the importance of solving a limit involving tanx and pi/4?

Solving a limit involving tanx and pi/4 is important because it helps us understand the behavior of the tangent function near the specific value of pi/4. This can be useful in various applications, such as calculating slopes and angles in trigonometry and physics problems.

## 4. Is there a specific method for solving a limit involving tanx and pi/4?

Yes, there are several methods that can be used to solve a limit involving tanx and pi/4, including the squeeze theorem, L'Hospital's rule, and trigonometric identities. The best method to use will depend on the specific function and limit being evaluated.

## 5. What are some common mistakes to avoid when solving a limit involving tanx and pi/4?

Some common mistakes to avoid when solving a limit involving tanx and pi/4 include forgetting to use trigonometric identities to rewrite the function, incorrectly applying L'Hospital's rule, and not checking for any potential discontinuities or undefined values at the specific value (pi/4). It is also important to carefully simplify the expression before plugging in the value to avoid any errors.

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