# How do you determine if #x/(x^2 -1)# is an even or odd function?

In our example:

graph{x/(x^2-1) [-10, 10, -5, 5]}

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To determine if the function ( \frac{x}{x^2 - 1} ) is even or odd, we evaluate ( f(-x) ) and compare it to ( f(x) ).

- If ( f(-x) = f(x) ) for all ( x ) in the domain, the function is even.
- If ( f(-x) = -f(x) ) for all ( x ) in the domain, the function is odd.
- If neither condition 1 nor 2 holds, the function is neither even nor odd.

Evaluating ( f(-x) ): [ f(-x) = \frac{-x}{(-x)^2 - 1} = \frac{-x}{x^2 - 1} ]

Comparing ( f(-x) ) with ( f(x) ): [ \frac{-x}{x^2 - 1} \neq \frac{x}{x^2 - 1} ]

Since ( f(-x) ) is not equal to ( f(x) ) and also not equal to the negative of ( f(x) ), the function ( \frac{x}{x^2 - 1} ) is neither even nor odd.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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