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The image contains two mathematical expressions to solve. Let's go through them one by one.

### Expression 1
\[
\lim_{{x \to 1}} \frac{x^3 - 1}{x - 1}
\]

This is a limit problem, and we can recognize the numerator \( x^3 - 1 \) as a difference of cubes. Recall the formula for the difference of cubes:

\[
a^3 - b^3 = (a-b)(a^2 + ab + b^2)
\]

Setting \( a = x \) and \( b = 1 \), we get:

\[
x^3 - 1 = (x-1)(x^2 + x + 1)
\]

Now substitute back into the limit expression:

\[
\lim_{{x \to 1}} \frac{(x-1)(x^2 + x + 1)}{x-1}
\]

The \( (x-1) \) terms cancel out:

\[
\lim_{{x \to 1}} (x^2 + x + 1)
\]

Now, substituting \( x = 1 \) directly into the simplified expression:

\[
1^2 + 1 + 1 = 3
\]

So, the value of the first limit is:

\[
\lim_{{x \to 1}} \frac{x^3 - 1}{x - 1} = 3
\]

### Expression 2
\[
\lim_{{n \to \infty}} \frac{n}{n+1} \left( 1 + \frac{1}{n} \right)^n
\]

To solve this limit, we can recognize that the expression \( \left(1 + \frac{1}{n}\right)^n \) resembles the definition of \( e \) as \( n \to \infty \):

\[
\lim_{{n \to \infty}} \left(1 + \frac{1}{n}\right)^n = e
\]

Thus, we rewrite the expression as:

\[
\lim_{{n \to \infty}} \frac{n}{n+1} \cdot e
\]

Now, simplify \( \frac{n}{n+1} \):

\[
\frac{n}{n+1} = \frac{n}{n(1 + \frac{1}{n})} = \frac{1}{1 + \frac{1}{n}}
\]

As \( n \to \infty \), \( \frac{1}{n} \to 0 \), so:

\[
\frac{1}{1 + \frac{1}{n}} \to 1
\]

Therefore, the entire limit becomes:

\[
\lim_{{n \to \infty}} \frac{n}{n+1} \cdot e = 1 \cdot e = e
\]

So, the value of the second limit is:

\[
\lim_{{n \to \infty}} \frac{n}{n+1} \left( 1 + \frac{1}{n} \right)^n = e
\]

### Summary
1. \(\lim_{{x \to 1}} \frac{x^3 - 1}{x - 1} = 3\)
2. \(\lim_{{n \to \infty}} \frac{n}{n+1} \left( 1 + \frac{1}{n} \right)^n = e\)

Would you like to go over any specific step in more detail, or do you have any questions?

### Related Questions
1. What is the formula for the difference of cubes?
2. How does the behavior of the function \( \left(1 + \frac{1}{n}\right)^n \) demonstrate the concept of \( e \)?
3. Why do we need to factor the numerator when solving the first limit?
4. What is the significance of the number \( e \) in calculus?
5. How can we use L'Hopital's rule to solve the first limit?

### Tip
When evaluating limits involving indeterminate forms, always look for algebraic simplifications or known limits (like the definition of \( e \)) to simplify the problem before applying more advanced methods like L'Hopital's rule.

Explore detailed solutions for limit problems in mathematics. Learn how to solve limits involving difference of cubes and exponential functions step-by-step. Suitable for undergraduate level study.

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