Function with denominator zero

In summary, The task is to find the limit of (tan 2x)/x as x approaches 0. To do this, we must first manipulate the equation so that the denominator does not equal zero. One method is to use L'Hopital's Rule, but another option is to use the variable substitution u=2x. It is also possible to use the "squeeze-play" theorem, as demonstrated with a similar example using tan(x). However, in this particular case, we must do some manipulation for the 2 in tan2x.
  • #1
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Homework Statement



This is a limit problem but what I need to figure out is simpler so I thought I'd post it under pre-calc. The question is:

Find the limit:

lim as x approaches 0 of (tan 2x)/x

Homework Equations





The Attempt at a Solution



Since x is in the denominator I know that I must re-write (tan 2x)/x so that the denominator doesn't equal zero. I also know that 0 is a root of both the numerator and denominator, but I don't know how to re-write such an equation. Any help? Thanks!
 
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  • #2
Use L'Hopital's Rule
 
  • #3
If you haven't done l'Hopital yet, do you know lim x->0 tan(x)/x=1 or lim x->0 sin(x)/x=1? Then you could just do the variable substitution u=2x.
 
  • #4
Why can't we obtain the limit here by squeezing?

I'll show you a similar example with tan(x) instead of (tan 2x):

[tex]\lim_{x \to 0} \frac{tanx}{x} = \lim_{x \to 0} (\frac{sin x}{x}.\frac{1}{cos x})[/tex]

[tex](\lim_{x \to 0} \frac{sin x}{x})(\lim_{x \to 0} \frac{1}{cos x}) = (1)(1) =1[/tex]

In your question you must do some manipulation for the 2 in tan2x.
 
  • #5
roam said:
Why can't we obtain the limit here by squeezing?

I'll show you a similar example with tan(x) instead of (tan 2x):

[tex]\lim_{x \to 0} \frac{tanx}{x} = \lim_{x \to 0} (\frac{sin x}{x}.\frac{1}{cos x})[/tex]

[tex](\lim_{x \to 0} \frac{sin x}{x})(\lim_{x \to 0} \frac{1}{cos x}) = (1)(1) =1[/tex]

In your question you must do some manipulation for the 2 in tan2x.
You're not actually doing any "squeezing" here--just using the fact that [tex]\lim_{x \to 0} \frac{sin x}{x} = 1[/tex]
This limit is often proved by the "squeeze-play" theorem, but can be done other ways.
 

1. What is a function with a denominator of zero?

A function with a denominator of zero is a mathematical expression where the denominator, or the number below the division line, is equal to zero. This can cause the function to be undefined or have an infinite value.

2. Can a function have a denominator of zero?

No, in general, a function cannot have a denominator of zero. This is because dividing by zero is undefined in mathematics and can lead to invalid or infinite results.

3. Why is dividing by zero not allowed in mathematics?

Dividing by zero is not allowed in mathematics because it leads to undefined or infinite results. It breaks the rules of arithmetic and can create inconsistencies in mathematical equations.

4. What happens when you try to evaluate a function with a denominator of zero?

When you try to evaluate a function with a denominator of zero, you will either get an error or an undefined result. This is because dividing by zero is not allowed in mathematics and cannot be solved.

5. Can a function with a denominator of zero have a limit?

Yes, a function with a denominator of zero can have a limit. This means that as the denominator approaches zero, the function will approach a specific value. However, the function will still be undefined at the point where the denominator is equal to zero.

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