If `f(a)=a^(2),phi(a)=b^(2)" and "f'(a)=n*phi'(a)," then: "underset(xtoa)("lim")(sqrt(f(x))-a)/(sqrtphi(x)-b)=`

Options

A. `(b)/(a)`
B. `n(b)/(a)`
C. `(a)/(b)`
D. `n(a)/(b)`

Answer

To solve the problem step by step, we need to find the limit:

\[ \lim_{x \to a} \frac{\sqrt{f(x)} - a}{\sqrt{\phi(x)} - b} \]

Given:
- \( f(a) = a^2 \)
- \( \phi(a) = b^2 \)
- \( f'(a) = n \cdot \phi'(a) \)

### Step 1: Substitute the values of \( f(x) \) and \( \phi(x) \)

We know that \( f(x) = x^2 \) and \( \phi(x) = y^2 \) for some function \( y \). Thus, we can write:

\[ \sqrt{f(x)} = \sqrt{x^2} = x \]
\[ \sqrt{\phi(x)} = \sqrt{y^2} = y \]

### Step 2: Rewrite the limit expression

Now substituting these into the limit expression:

\[ \lim_{x \to a} \frac{x - a}{y - b} \]

### Step 3: Evaluate the limit directly

When we substitute \( x = a \) and \( y = b \), we get:

\[ \frac{a - a}{b - b} = \frac{0}{0} \]

This is an indeterminate form, so we can apply L'Hôpital's Rule.

### Step 4: Apply L'Hôpital's Rule

Using L'Hôpital's Rule, we differentiate the numerator and the denominator:

- The derivative of the numerator \( x - a \) with respect to \( x \) is \( 1 \).
- The derivative of the denominator \( y - b \) with respect to \( x \) is \( \frac{dy}{dx} \).

Thus, we have:

\[ \lim_{x \to a} \frac{1}{\frac{dy}{dx}} \]

### Step 5: Find \( \frac{dy}{dx} \)

Using the chain rule, we know that:

\[ \frac{dy}{dx} = \phi'(x) = \frac{d}{dx}(y^2) = 2y \cdot \frac{dy}{dx} \]

From the problem, we know that \( f'(a) = n \cdot \phi'(a) \). Hence, we can express this as:

\[ f'(a) = 2a \quad \text{and} \quad \phi'(a) = 2b \]

Thus,

\[ 2a = n \cdot 2b \implies n = \frac{a}{b} \]

### Step 6: Substitute back into the limit

Now substituting back into our limit expression:

\[ \lim_{x \to a} \frac{1}{\frac{dy}{dx}} = \frac{1}{\phi'(a)} = \frac{1}{2b} \]

### Step 7: Final limit calculation

Now substituting \( n = \frac{a}{b} \):

\[ \lim_{x \to a} \frac{\sqrt{f(x)} - a}{\sqrt{\phi(x)} - b} = \frac{n \cdot b}{a} = \frac{a}{b} \cdot \frac{b}{a} = n \]

### Conclusion

Thus, the final answer for the limit is:

\[ \frac{n \cdot b}{a} \]