Let F(x) = ∫0 to x e^(t^2)dt
a) Compute limx → ∞ (xF(x))/ (e^(x^2))
b) Compute limx → 0 F(x)/(xe^(x^2))
To compute the limits, we need to evaluate the given expressions and apply the limit operation as x approaches the specified values.
a) Compute limx → ∞ (xF(x))/ (e^(x^2)):
To evaluate this limit, we start by computing F(x):
F(x) = ∫0 to x e^(t^2)dt
Unfortunately, there is no simple closed-form expression for ∫e^(t^2)dt, which means we cannot evaluate F(x) directly.
However, we can approximate the limit by using numerical methods or evaluating the expression for large values of x. Let's use numerical integration to approximate F(x) for a large value of x:
1. Divide the interval [0, x] into smaller subintervals.
2. Approximate the area under the curve e^(t^2) using a numerical integration method such as Simpson's rule or the trapezoidal rule.
3. Sum the areas of all subintervals to obtain the approximate value of F(x).
For example, let's approximate F(10) using Simpson's rule with 10 subintervals:
Δx = (10 - 0) / 10 = 1
F(10) ≈ Δx / 3 * [f(0) + 4f(1) + 2f(2) + 4f(3) + 2f(4) + 4f(5) + 2f(6) + 4f(7) + 2f(8) + 4f(9) + f(10)]
where f(x) = e^(x^2).
Plugging in the values:
F(10) ≈ (1 / 3) * [e^(0) + 4e^(1) + 2e^(4) + 4e^(9) + 2e^(16) + 4e^(25) + 2e^(36) + 4e^(49) + 2e^(64) + 4e^(81) + e^(100)]
Approximating this expression will give us an estimate of F(10). Repeat this process with larger values of x to refine the approximation.
Once we have the approximation for F(x), we can find the limit by evaluating the expression:
limx → ∞ (xF(x))/ (e^(x^2))
Substitute the obtained approximation for F(x) and evaluate the expression as x approaches infinity.
b) Compute limx → 0 F(x)/(xe^(x^2)):
For this limit, we can use L'Hôpital's rule, which states that if we have an indeterminate form of the type 0/0 or ∞/∞, we can take the derivative of the numerator and denominator repeatedly until we no longer have an indeterminate form. Let's use this rule to evaluate the limit.
First, let's differentiate F(x) with respect to x:
F'(x) = (d/dx) ∫0 to x e^(t^2)dt
Using the Fundamental Theorem of Calculus, we have:
F'(x) = e^(x^2)
Now, let's differentiate the numerator and denominator of the given expression:
limx → 0 F(x)/(xe^(x^2)) = limx → 0 F'(x)/(e^(x^2) + 2x * e^(x^2))
= limx → 0 e^(x^2)/(e^(x^2) + 2x * e^(x^2))
= limx → 0 1 / (1 + 2x)
Substituting x = 0 into the expression will give us the value of the limit:
limx → 0 1 / (1 + 2x) = 1 / (1 + 2 * 0) = 1.