Find the limits in Exercises 59-62 lim((1)/(x^2 / 3)+(2)/((x-1)^2 / 3)) as a. x → 0^+

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Find the limits in Exercises 59-62 lim((1)/(x^2 /...

Find the limits in Exercises $59-62$ $$\lim \left(\frac{1}{x^{2 / 3}}+\frac{2}{(x-1)^{2 / 3}}\right) as$$ \begin{equation} \begin{array}{ll}{\text { a. }} & {x \rightarrow 0^{+}} & {\text { b. } x \rightarrow 0^{-}} \\ {\text { c. }} & {x \rightarrow 1^{+}} & {\text { d. } x \rightarrow 1^{-}}\end{array} \end{equation}

Find the limits in Exercises $59-62$
$$\lim \left(\frac{1}{x^{2 / 3}}+\frac{2}{(x-1)^{2 / 3}}\right) as$$
\begin{equation}
\begin{array}{ll}{\text { a. }} & {x \rightarrow 0^{+}} & {\text { b. } x \rightarrow 0^{-}} \\ {\text { c. }} & {x \rightarrow 1^{+}} & {\text { d. } x \rightarrow 1^{-}}\end{array}
\end{equation}

Key Concepts

Singularities and Asymptotic Behavior

Singularities occur at points where a function is undefined or unbounded, often leading to values of infinity in the context of limits. Recognizing and analyzing these points is crucial in understanding the asymptotic behavior of the function, particularly when calculating limits that approach these problematic values. Such analysis may reveal infinite limits, jumps or discontinuities that are significant in the overall understanding of the function's continuity and differentiability.

Fractional Exponents

Fractional exponents, such as exponent 2/3, generalize the idea of powers and roots. They are key in expressing functions that involve radicals in an algebraically convenient form. Understanding fractional exponents is important when evaluating limits because they affect the function's growth and behavior near singularities or points where the function may not be defined in the usual manner.

One-sided Limit

A one-sided limit considers the behavior of a function as the variable approaches a point from one side—either from the left (negative side) or the right (positive side). This notion is essential in handling cases where the function's behavior differs depending on the side of approach, which is particularly useful when studying discontinuities or singularities.

Limit

A limit examines the behavior of a function as the input approaches a particular value. It is a fundamental concept in calculus and real analysis, used to analyze functions near points of interest, especially where the function may be undefined or exhibit unusual behavior. The limit concept underlies the definitions of continuity, derivatives, and integrals.

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