`lim_(x to a) (x^(m)-a^(m))/(x^(n)-a^(n))` is equal to

Options

A. ` mna^(m-n)`
B. `m/na^(m-n)`
C. `n/ma^(m-n)`
D. `mna^(m+a)`

Answer

To solve the limit

\[ \lim_{x \to a} \frac{x^m - a^m}{x^n - a^n}, \]

we first observe that substituting \( x = a \) directly into the expression results in the indeterminate form \( \frac{0}{0} \). Therefore, we can apply L'Hôpital's Rule, which states that if the limit results in an indeterminate form, we can differentiate the numerator and the denominator separately and then take the limit again.

### Step 1: Differentiate the Numerator and Denominator

Differentiate the numerator \( x^m - a^m \):
\[ \frac{d}{dx}(x^m - a^m) = mx^{m-1}. \]

Differentiate the denominator \( x^n - a^n \):
\[ \frac{d}{dx}(x^n - a^n) = nx^{n-1}. \]

### Step 2: Apply L'Hôpital's Rule

Now we can apply L'Hôpital's Rule:
\[ \lim_{x \to a} \frac{x^m - a^m}{x^n - a^n} = \lim_{x \to a} \frac{mx^{m-1}}{nx^{n-1}}. \]

### Step 3: Substitute \( x = a \)

Now substitute \( x = a \) into the limit:
\[ \frac{m a^{m-1}}{n a^{n-1}}. \]

### Step 4: Simplify the Expression

This simplifies to:
\[ \frac{m}{n} \cdot \frac{a^{m-1}}{a^{n-1}} = \frac{m}{n} a^{(m-1) - (n-1)} = \frac{m}{n} a^{m-n}. \]

### Final Answer

Thus, the limit is:
\[ \lim_{x \to a} \frac{x^m - a^m}{x^n - a^n} = \frac{m}{n} a^{m-n}. \]

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