find the tangent function is continuous on the interval (-90,90) but is not uniformly continuous
1/(x^2-90^2)
To determine whether the tangent function is continuous on the interval (-90, 90), we need to check two things: if the function is defined for all values on the interval and if it has no abrupt changes or discontinuities.
First, let's consider whether the tangent function is defined for all values in the interval (-90, 90). The tangent function is defined for all values except those where the cosine function (denominator of tangent) becomes zero. Cosine is zero at -90°, 90°, and multiples of 180°. However, since the interval specified does not include -90° and 90°, the tangent function is defined for all values in (-90, 90).
Next, let's examine whether the tangent function has any abrupt changes or discontinuities. The tangent function has vertical asymptotes at -90° and 90°, which means it approaches infinity near these points. Although it has discontinuities at these points, since they are not included in the interval (-90, 90), the tangent function does not have any abrupt changes or discontinuities within this interval.
Hence, we can conclude that the tangent function is continuous on the interval (-90, 90).
However, the tangent function is not uniformly continuous on (-90, 90). Uniform continuity requires that no matter how small the chosen interval ε, there is a corresponding δ such that for any two points x and y within the interval (-90, 90) such that |x - y| < δ, we have |tan(x) - tan(y)| < ε.
In the case of the tangent function, as we approach the vertical asymptotes at -90° and 90°, the rate at which tangent values change becomes significantly steep. Hence, no matter how small the ε value is chosen, we cannot find a corresponding δ that works uniformly for the entire interval (-90, 90).
Therefore, the tangent function is continuous but not uniformly continuous on the interval (-90, 90).