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Let's work through the problem step by step, based on the image you have uploaded:

### (a) Initial Limit Expression

The given limit to evaluate is:

\[
\lim_{x \to 1} \frac{x^{1505} - 1}{x - 1}.
\]

This limit looks challenging to evaluate directly by substitution because if we substitute \(x = 1\), we get a \(\frac{0}{0}\) indeterminate form. Therefore, we need a different approach.

### (b) Substitution \( h = x - 1 \)

We are asked to make the substitution \( h = x - 1 \). Let's rewrite the limit in terms of \(h\). Notice that if \( x \to 1 \), then \( h \to 0 \). Hence:

\[
\lim_{x \to 1} \frac{x^{1505} - 1}{x - 1} = \lim_{h \to 0} \frac{(h + 1)^{1505} - 1}{h}.
\]

This is the transformed limit using the substitution \( h = x - 1 \). We will simplify this expression in the next step.

### (c) Derivative Interpretation

The expression we have from part (b) looks similar to the definition of a derivative. In fact, we can express the limit as the derivative of some function \( f(x) \) at a point \( a \).

We recognize that this is the derivative of the function \( f(x) = x^{1505} \) evaluated at \( a = 1 \). This is because:

\[
f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}.
\]

Here, \( f(x) = x^{1505} \), so the derivative of \( f(x) \) is:

\[
f'(x) = 1505 x^{1504}.
\]

Thus, the original limit is:

\[
f'(1) = 1505 \cdot 1^{1504} = 1505.
\]

### Final Answer

Therefore, the value of the limit is:

\[
\lim_{x \to 1} \frac{x^{1505} - 1}{x - 1} = 1505.
\]

---

Would you like further details on any of the steps? Here are some related questions to consider:

1. Why does the substitution \( h = x - 1 \) help simplify the limit?
2. How does this problem relate to the definition of a derivative?
3. Can L'Hôpital's rule be applied to this limit? If so, how would the process compare?
4. What other methods could be used to evaluate limits like this?
5. How can you use the binomial expansion to solve the limit without calculus?

**Tip:** Substituting variables often transforms limits into easier forms, making them approachable by using known calculus tools like the derivative definition.

Evaluate the limit lim_{x→1} [(x^1505 − 1) / (x − 1)] and perform substitution h = x − 1.

Learn how to evaluate the limit lim_{x→1} [(x^1505 − 1) / (x − 1)] by applying substitution and interpreting the expression as a derivative. This problem involves key calculus concepts such as limits, substitution techniques, and the power rule for derivatives.

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