Question: How do you compute the following limits in a small amount of time?

$$\lim_{(a,b,c,d)\to (-\infty,0^{-},0^{+},+\infty)}\frac{\int_{a}^{b}1/x+\int_{c}^{d}1/x}{(b-a)+(d-c)}$$

$$\lim_{(a,b,c,d)\to (-\infty,0^{-},0^{+},+\infty)}\frac{\int_{a}^{b}1/(x^{2/3})+\int_{c}^{d}1/(x^{2/3})}{(b-a)+(d-c)}$$

Attempt: I tried the following:

Limit[(1/((b - a) + (d - c))) (Integrate[1/x, {x, a, b}] + 
    Integrate[1/x, {x, c, d}]), {a, b, c, d} -> {-Infinity, 0,
    0, Infinity}, Direction -> {Reals, -1, 1, Reals}]

Limit[(1/((b - a) + (d - c))) (Integrate[1/(x^(2/3)), {x, a, b}] + 
    Integrate[1/(x^(2/3)), {x, c, d}]), {a, b, c, d} -> {-Infinity, 0,
    0, Infinity}, Direction -> {Reals, -1, 1, Reals}]

However, the computation time of the limits are too long. What can be made to shorten the time?

Motivation: Let $n\in\mathbb{N}$ and suppose $f:A\subseteq\mathbb{R}^{n}\to\mathbb{R}$ is a function, where $A$ and $f$ are Borel. I want an example of an explicit, unbounded, and continuous $f$ such that the extended mean, w.r.t. the Hausdorff measure in its dimension, is finite. Here is the definition of the extended mean (see Definition 12 [pg. 11-12] of the attatchment):

Definition (The Extended Mean of $f$)

Suppose:

• $|\cdot|$ is the absolute value
• $\dim_{\mathrm{H}}(\cdot)$ is the Hausdorff dimension
• $\mathcal{H}^{\dim_{\mathrm{H}}(\cdot)}(\cdot)$ is the Hausdorff measure in its dimension on the Borel $\sigma$-algebra
• the integral is defined, w.r.t.\ the Hausdorff measure in its dimension
• $\mathcal{A}(A)$ is the index set
• $\{C_{\gamma}:\gamma\in\mathcal{A}(A)\}$ is a family of sets
• $A$ is a Borel subset of $\mathbb{R}^n$
• $\mathbb{B}(A)$ is the set of all families of bounded sets with set-theoretic limit $A$ such that $\{C_{\gamma}:\gamma\in\mathcal{A}(A)\}\in\mathbb{B}(A)$, when $\dim_{\mathrm{H}}(C_{\gamma})=\dim_{\mathrm{H}}(A)$ for all $\gamma\in\mathcal{A}(A)$

The extended mean of $f:A\subseteq\mathbb{R}^{n}\to\mathbb{R}$ (i.e., $f$ is Borel), w.r.t.\ the Hausdorff measure in its dimension, is $M_{f}(A;n)$ when the following is true:

$$\small{\begin{align}& \forall(\{A_r:r\in\mathcal{A}(A)\}\in\mathbb{B}(A))\exists!(M_{f}(A;n)\in\mathbb{R})\forall(\epsilon>0)\exists(N\in\mathcal{A}(A))\forall(r\in\mathcal{A}(A))\nonumber\\ &\left(r\ge N\Rightarrow\left|\frac{1}{{\mathcal{H}}^{\dim_{\mathrm{H}}(A_r)}(A_r)}\int_{A_r}f\, d{\mathcal{H}}^{\dim_{\mathrm{H}}(A_r)}-M_{f}(A;n)\right|<\epsilon\right) \end{align}}$$

and when no such $M_{f}(A;n)$ exists, $M_f(A;n)$ is infinite or undefined.

Attachments:

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