Evaluate the following limits :
`Lim_( x to 5) (x^(4) - 625 )/(x^(3) - 125)`

Answer

To evaluate the limit

\[ \lim_{x \to 5} \frac{x^4 - 625}{x^3 - 125}, \]

we will follow these steps:

### Step 1: Substitute the limit value
First, we substitute \( x = 5 \) into the expression:

\[ \frac{5^4 - 625}{5^3 - 125} = \frac{625 - 625}{125 - 125} = \frac{0}{0}. \]

This results in an indeterminate form \( \frac{0}{0} \).

**Hint:** When you encounter a \( \frac{0}{0} \) form, you can use L'Hôpital's Rule.

### Step 2: Apply L'Hôpital's Rule
Since we have an indeterminate form, we can apply L'Hôpital's Rule, which states that:

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

provided the limit on the right side exists.

Here, let \( f(x) = x^4 - 625 \) and \( g(x) = x^3 - 125 \).

**Hint:** Differentiate the numerator and denominator separately.

### Step 3: Differentiate the numerator and denominator
Now we differentiate \( f(x) \) and \( g(x) \):

- The derivative of \( f(x) = x^4 - 625 \) is \( f'(x) = 4x^3 \).
- The derivative of \( g(x) = x^3 - 125 \) is \( g'(x) = 3x^2 \).

**Hint:** Remember to differentiate each term separately.

### Step 4: Rewrite the limit using derivatives
Now we can rewrite the limit using the derivatives:

\[ \lim_{x \to 5} \frac{f'(x)}{g'(x)} = \lim_{x \to 5} \frac{4x^3}{3x^2}. \]

**Hint:** Simplify the expression before substituting the limit value.

### Step 5: Simplify the expression
We can simplify the expression:

\[ \frac{4x^3}{3x^2} = \frac{4}{3} x. \]

**Hint:** Now substitute \( x = 5 \) into the simplified expression.

### Step 6: Substitute \( x = 5 \)
Now we substitute \( x = 5 \):

\[ \lim_{x \to 5} \frac{4}{3} x = \frac{4}{3} \cdot 5 = \frac{20}{3}. \]

### Final Answer
Thus, the limit is

\[ \boxed{\frac{20}{3}}. \]