Calculating Limit as x-->pi/4: Solve Tan(x-pi/4)+1/x-pi/4

In summary, the given limit is of the form "1/0" and thus does not exist. There is no way to change the limit into a derivative-limit, as a derivative would yield "0/0" instead of "1/0".
  • #1
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Homework Statement



Calculate the limit as x-->pi/4 of [tan(x-pi/4)+1]/x-pi/4

Homework Equations



lim h-->0 of [f(x+h)-f(x)]/h = f'(x)

lim x--->a [f(x)-f(a)]/(x-a) = f'(x)

The Attempt at a Solution



I've attempted to turn this equation into the form f(x)-f(a)/x-a by letting f(x)=tanx and a=pi/4

This turns into -[-tan(x+a)-tan(a)]/x-a...which isn't the correct derivative form. .I've tried other methods which also turn into similar garble (a minus sign backwards, x-h rather than x+h and the like).

Can anyone see what the problem is? Thanks
 
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  • #2
The limit

[tex] \lim_{x\rightarrow \pi/4}{\frac{\tan(x-\pi/4)+1}{x-\pi/4}} [/tex]

Is of the form "1/0". Thus the limit is always [tex]+\infty[/tex] or [tex]-\infty [/tex] or it doesn't exist (if the left limit does not equal the right limit). Which one is it?
 
  • #3
Oh is is actually this easy?

In that case, I would think since the numerator will always be positive regardless of which side the limit approaches from, and the denominator will switch signs depending on which side it approaches from, then the limit from the left will be -infinity and will be +infinite from the right, then the limit will not exist.

Are you sure there isn't a way to do this with derivatives? I thought this is what the question was getting at
 
  • #4
Yes, I know it looks a lot like a derivative. But this method is definitely simpler then to change the limit into a derivative (if there is a way of doing that).

I don't think that you can change this limit into a derivative-limit. A derivative will yield "0/0", while this is "1/0".
 
  • #5
Well, if the question was

[tex]\lim_{x\rightarrow \pi/4}{\frac{\tan(x)-1}{x-\pi/4}} [/tex]

Then you can do some derivative-stuff. But I don't really see a possibility here...
 
  • #6
Okay that makes sense. Thanks for the help
 

What is a limit?

A limit is a fundamental concept in calculus that represents the value that a function approaches as its input (x) approaches a specified point or value.

Why is it important to calculate limits?

Calculating limits allows us to understand the behavior of a function near a particular point and determine important properties such as continuity and differentiability.

How do I calculate the limit of a function?

To calculate the limit of a function, we evaluate the function at values of x that are very close to the specified point and observe the trend in the output values. We can also use algebraic techniques and calculus rules to simplify the function and determine the limit.

What does x->pi/4 mean in the context of this problem?

In this problem, x->pi/4 represents the input (x) approaching the value of pi/4. This is the point at which we are trying to calculate the limit of the function.

Can we use a calculator to calculate the limit of this function?

Yes, we can use a calculator to calculate the limit of this function. Most scientific calculators have a "limit" function that allows us to input a function and a value to approach, such as x->pi/4, and it will give us the resulting limit value.

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