https://file.aiquickdraw.com/mathbot/file/1728033444904fy4csivp.jpg

The expression in the image involves finding the limit as \( x \to \infty \) of the following function:

\[
\lim_{x \to \infty} \frac{x \cdot \left( x^{1/x} - 1 \right)}{Lx}
\]

We can simplify and solve this limit step by step. Here's the breakdown:

### Step 1: Simplify the expression
First, notice that the denominator is simply \( Lx \), where \( L \) is a constant.

Now, consider the term \( x^{1/x} \). We know from limits that:

\[
\lim_{x \to \infty} x^{1/x} = 1
\]

This can be shown using logarithms and L'Hopital's Rule, but it's a known result.

So, for large \( x \):

\[
x^{1/x} - 1 \approx 0
\]

### Step 2: Approximate the behavior of \( x^{1/x} - 1 \)
To get a better approximation for large \( x \), we can use a first-order Taylor expansion around \( x \to \infty \). Using this, we get:

\[
x^{1/x} - 1 \approx \frac{1}{x}
\]

### Step 3: Substitute the approximation
Now, substitute this approximation into the original expression:

\[
\frac{x \cdot \left( x^{1/x} - 1 \right)}{Lx} \approx \frac{x \cdot \frac{1}{x}}{Lx} = \frac{1}{L}
\]

### Step 4: Evaluate the limit
Thus, as \( x \to \infty \), the limit becomes:

\[
\lim_{x \to \infty} \frac{x \cdot \left( x^{1/x} - 1 \right)}{Lx} = \frac{1}{L}
\]

So, the solution to the limit is \( \frac{1}{L} \).

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Do you need more details or have any questions?

### Related questions:
1. How can Taylor expansions be used to approximate functions near infinity?
2. What is L'Hopital's rule, and when is it useful for limits?
3. How can logarithmic differentiation help in solving limits involving powers?
4. How does the behavior of \( x^{1/x} \) change as \( x \) tends to infinity?
5. Can you generalize this problem for other functions of the form \( x^{f(x)} \)?

### Tip:
When evaluating limits involving expressions like \( x^{1/x} \), it helps to know common limits and expansions to quickly approximate the behavior of terms near infinity.

Find the limit of the expression lim (x -> ∞) (x * (x^(1/x) - 1)) / (Lx).

Learn how to compute the limit lim (x -> ∞) (x * (x^(1/x) - 1)) / (Lx) using asymptotic approximations and Taylor expansions. This problem covers key calculus concepts, including limits and L'Hopital's Rule. Suitable for advanced high school or first-year university students.

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