Question

We define the function $\qquad f(x) \triangleq \frac{\sin(x)}{x}$, \quad for $x \in (0, \infty)$. 1. Compute the limit $\lim_{x \downarrow 0} f(x)$. We say a function $g: \mathbb{R} \to \mathbb{R}$ is (Lebesgue) $p$-integrable and write $g \in L^p$ if we have $\qquad \int_{\mathbb{R}} |g(x)|^p dx < \infty$, $p \in [1, \infty)$. When the above is true for $p = 1$ we simply say $g$ is integrable. 2. Prove that $f$ defined above is not integrable over $(0, \infty)$. 3. Prove that $f$ is integrable in the improper Riemann sense by computing the limit $\qquad \lim_{a \to \infty} \int_0^a f(x) dx$. Hint: Link the above integral with the double integral of the function $e^{-xy} \sin(x)$ for $0 < x < a$, $y > 0$. Then compute it by switching the order of integration

          We define the function
$\qquad f(x) \triangleq \frac{\sin(x)}{x}$, \quad for $x \in (0, \infty)$.

1. Compute the limit $\lim_{x \downarrow 0} f(x)$.
We say a function $g: \mathbb{R} \to \mathbb{R}$ is (Lebesgue) $p$-integrable and write $g \in L^p$ if we have
$\qquad \int_{\mathbb{R}} |g(x)|^p dx < \infty$, $p \in [1, \infty)$.
When the above is true for $p = 1$ we simply say $g$ is integrable.

2. Prove that $f$ defined above is not integrable over $(0, \infty)$.

3. Prove that $f$ is integrable in the improper Riemann sense by computing the limit
$\qquad \lim_{a \to \infty} \int_0^a f(x) dx$.

Hint: Link the above integral with the double integral of the function $e^{-xy} \sin(x)$ for $0 < x < a$, $y > 0$. Then compute it by switching the order of integration

We define the function
f(x) ≜(sin(x))/(x), for x ∈ (0, ∞).

1. Compute the limit limx ↓ 0 f(x).
We say a function g: ℝ→ℝ is (Lebesgue) p-integrable and write g ∈ L^p if we have
∫ℝ |g(x)|^p dx < ∞, p ∈ [1, ∞).
When the above is true for p = 1 we simply say g is integrable.

2. Prove that f defined above is not integrable over (0, ∞).

3. Prove that f is integrable in the improper Riemann sense by computing the limit
lima →∞∫0^a f(x) dx.

Hint: Link the above integral with the double integral of the function e^-xysin(x) for 0 < x < a, y > 0. Then compute it by switching the order of integration

Added by Nicholas R.

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