find the x-values at which f is not continuous. which of the discontinuities are removable?
f(x)=1/(x^2+1)
To find the x-values at which a function is not continuous, we need to identify any points where the function exhibits any of the three types of discontinuities: removable, jump, or infinite discontinuity.
Let's analyze the function f(x) = 1/(x^2 + 1) to determine its discontinuities.
1. Removable Discontinuity:
A removable discontinuity occurs when a function has a hole at a certain x-value. To identify removable discontinuities, we need to check if the function can be redefined at that point to make it continuous.
In this case, f(x) = 1/(x^2 + 1) is a rational function, so it is continuous except where the denominator is equal to zero.
To find the x-values that make the denominator zero, we solve the equation x^2 + 1 = 0:
x^2 = -1
Since we cannot take the square root of a negative number in the real number system, this equation has no real solutions. Therefore, there are no removable discontinuities for f(x).
2. Jump Discontinuity:
A jump discontinuity occurs when the function has two distinct finite limits as x approaches a certain value from the left and the right.
In this case, since f(x) = 1/(x^2 + 1), the function does not have any jump discontinuities because there are no sudden jumps in the function's values.
3. Infinite Discontinuity:
An infinite discontinuity occurs when the function has an asymptote as x approaches a certain value from the left or the right.
In this case, f(x) = 1/(x^2 + 1) does have infinite discontinuities. As x approaches positive or negative infinity, the denominator (x^2 + 1) approaches infinity while the numerator remains finite. Therefore, the function approaches zero as x approaches infinity or negative infinity.
In summary, the function f(x) = 1/(x^2 + 1) has no removable discontinuities, no jump discontinuities, and has infinite discontinuities as x approaches positive or negative infinity.
the function is not continuous when the denominator is zero, or...
0 = x^2+1
solving for x then
x = i
It is a removable discontinuity since it can be repaired by replacing a single point.