Evaluate limit of x approaches to 5 and the function is (x^7-5^7)/(x-5)

To evaluate the limit of the function \(\frac{x^7 - 5^7}{x - 5}\) as \(x\) approaches 5, we can use the limit properties and algebraic manipulation. The expression \(\frac{x^7 - 5^7}{x - 5}\) takes the indeterminate form \(\frac{0}{0}\) when \(x = 5\). Therefore, we can apply L'Hôpital's Rule, which states that if \(\lim_{x \to c} \frac{f(x)}{g(x)}\) is of the form \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\), then

\[
\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}
\]

Given:
\[ f(x) = x^7 - 5^7 \]
\[ g(x) = x - 5 \]

First, we compute the derivatives \(f'(x)\) and \(g'(x)\):

\[ f'(x) = \frac{d}{dx}(x^7 - 5^7) = 7x^6 \]
\[ g'(x) = \frac{d}{dx}(x - 5) = 1 \]

Now, applying L'Hôpital's Rule:

\[
\lim_{x \to 5} \frac{x^7 - 5^7}{x - 5} =

Learn how to evaluate the limit of the function (x^7 - 5^7) / (x - 5) as x approaches 5 using L'Hôpital's Rule and algebraic manipulation. This problem involves advanced concepts in calculus, including derivatives and limits. Suitable for advanced high school students and college-level calculus learners.

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