To find the value of [tex]\( k \)[/tex] such that the given limit holds true, let's solve the problem step-by-step.
Step 1: Set Up the Integral
The problem requires evaluating the limit of the integral as [tex]\( t \)[/tex] approaches [tex]\( +\infty \)[/tex]:
[tex]\[ \lim_{t \to \infty} \int_2^t \frac{k}{x^2} \, dx = 1 \][/tex]
Step 2: Compute the Definite Integral
Let's find the definite integral [tex]\( \int_2^t \frac{k}{x^2} \, dx \)[/tex].
We start by integrating [tex]\( \frac{k}{x^2} \)[/tex]. Recall that the integral of [tex]\( x^{-2} \)[/tex] is [tex]\( -x^{-1} \)[/tex], so we have:
[tex]\[ \int \frac{k}{x^2} \, dx = \int k \cdot x^{-2} \, dx = k \left( -\frac{1}{x} \right) = -\frac{k}{x} \][/tex]
Now we need to evaluate this from [tex]\( 2 \)[/tex] to [tex]\( t \)[/tex]:
[tex]\[ \int_2^t \frac{k}{x^2} \, dx = \left[ -\frac{k}{x} \right]_2^t \][/tex]
Step 3: Evaluate the Definite Integral
Substitute the limits of integration:
[tex]\[ \left[ -\frac{k}{x} \right]_2^t = -\frac{k}{t} - \left(-\frac{k}{2}\right) = -\frac{k}{t} + \frac{k}{2} = \frac{k}{2} - \frac{k}{t} \][/tex]
So the integral evaluates to:
[tex]\[ \int_2^t \frac{k}{x^2} \, dx = \frac{k}{2} - \frac{k}{t} \][/tex]
Step 4: Take the Limit as [tex]\( t \)[/tex] Approaches Infinity
We need to find:
[tex]\[ \lim_{t \to \infty} \left( \frac{k}{2} - \frac{k}{t} \right) \][/tex]
As [tex]\( t \)[/tex] approaches infinity, [tex]\( \frac{k}{t} \)[/tex] approaches 0. Hence, we have:
[tex]\[ \lim_{t \to \infty} \left( \frac{k}{2} - \frac{k}{t} \right) = \frac{k}{2} \][/tex]
Step 5: Set the Limit Equal to 1 and Solve for [tex]\( k \)[/tex]
According to the problem, the limit should equal 1:
[tex]\[ \frac{k}{2} = 1 \][/tex]
To solve for [tex]\( k \)[/tex], multiply both sides by 2:
[tex]\[ k = 2 \][/tex]
Conclusion:
Therefore, the value of [tex]\( k \)[/tex] that satisfies the given limit is [tex]\( k = 2 \)[/tex]. The integral expression, the limit, and the solved value of [tex]\( k \)[/tex] can then be summarized as:
- Integral expression: [tex]\( \frac{k}{2} - \frac{k}{t} \)[/tex]
- Limit as [tex]\( t \)[/tex] approaches infinity: [tex]\( \frac{k}{2} \)[/tex]
- Value of [tex]\( k \)[/tex]: [tex]\( k = 2 \)[/tex]
