# How do you find the x values at which #f(x)=x/(x^2+1)# is not continuous, which of the discontinuities are removable?

is a rational function, that is the quotient of two polynomials. As such it is defined and continuous everywhere except where the denominator vanishes.

However as:

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To find the x values at which f(x) = x/(x^2+1) is not continuous, we need to identify the points where the function is undefined or where the limit does not exist.

The function f(x) is undefined when the denominator, x^2+1, equals zero. Solving x^2+1 = 0, we find that x^2 = -1, which has no real solutions. Therefore, the function is defined for all real values of x.

To determine if there are any points where the limit does not exist, we need to check the behavior of the function as x approaches certain values.

Taking the limit as x approaches positive or negative infinity, we find that the function approaches zero. Therefore, there are no vertical asymptotes or infinite discontinuities.

Next, we check for any removable discontinuities. A removable discontinuity occurs when the function is defined at a certain point but has a hole or gap in the graph. To identify these points, we need to check if the function can be simplified or if there are any common factors that can be canceled out.

Simplifying f(x) = x/(x^2+1), we cannot cancel out any common factors. Therefore, there are no removable discontinuities in this function.

In conclusion, the function f(x) = x/(x^2+1) is continuous for all real values of x, and there are no removable discontinuities.

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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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