Let f(x)=|x|^x, x ≠0 . Show that f has a removable discontinuity at x=0 and extend the definitio

step 1 Suppose f of x is equal to the absolute value of x raised to x for x not equal to zero. Now for

Let f(x)=|x|^x, x ≠0 . Show that f has a removable...

Key Concepts

Removable Discontinuity

A removable discontinuity occurs at a point in the domain of a function when the limit exists at that point, but the function is either not defined there or its defined value does not equal the limit. This type of discontinuity can be 'removed' by redefining the function at the critical point so that it becomes continuous, highlighting the importance of understanding the intrinsic limit properties of the function.

Limit Evaluation

The evaluation of limits is a fundamental concept in calculus that determines the behavior of a function as its variable approaches a specific value. In problems involving continuity or discontinuities, computing the limit allows one to ascertain what the value of the function would be at the point if it were defined, thus guiding the appropriate way to reassign a value for continuity.

Continuous Extension

A continuous extension of a function involves redefining the function at points where it was originally undefined or discontinuous in order to ensure overall continuity. By assigning values based on the computed limits at the discontinuous points, the modified function maintains the key property that small changes in input yield small changes in output across its entire domain.

Exponential and Logarithmic Manipulation

Exponential functions and their properties, often analyzed via logarithmic transformations, play a major role in limit computations, especially when the function involves exponentiation with variable exponents. Converting exponential expressions to a logarithmic form can simplify the evaluation process by enabling the use of limit laws and approximation techniques, which is essential in understanding behaviors near points of discontinuity.

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