To solve the problem, we need to find the limit of $y = f(x) = k \left( \frac{a^2 x^2 - b^2}{ax - b} \right)$ as $x \to \frac{b}{a}$, given the constants $a = 7.2$, $b = 3.9$, and $k = 3.2$.

Step 1: Substitute $x = \frac{b}{a}$

Substitute $a = 7.2$ and $b = 3.9$ to get $x = \frac{b}{a} = \frac{3.9}{7.2} \approx 0.5417$.

Step 2: Simplify the Expression

Substitute $x = \frac{b}{a}$ into the expression for $f(x)$: $f\left(\frac{b}{a}\right) = k \left( \frac{a^2 \left(\frac{b}{a}\right)^2 - b^2}{a \left(\frac{b}{a}\right) - b} \right).$

Step 3: Simplify the Numerator and Denominator

• Numerator: $a^2 \left(\frac{b}{a}\right)^2 = \frac{a^2 b^2}{a^2} = b^2$, so $a^2 \left(\frac{b}{a}\right)^2 - b^2 = b^2 - b^2 = 0$.

• Denominator: $a \left(\frac{b}{a}\right) - b = b - b = 0$.

This leads to an indeterminate form $\frac{0}{0}$, so we need to use L'Hôpital's Rule.

Step 4: Apply L'Hôpital's Rule

To apply L'Hôpital's Rule, find the derivatives of the numerator and the denominator:

• Derivative of the numerator $a^2 x^2 - b^2$ with respect to $x$ is $2a^2 x$.

• Derivative of the denominator $ax - b$ with respect to $x$ is $a$.

So, applying L'Hôpital's Rule: $\lim_{x \to \frac{b}{a}} f(x) = k \lim_{x \to \frac{b}{a}} \frac{2a^2 x}{a} = k \cdot 2a \cdot \frac{b}{a} = k \cdot 2b.$

Step 5: Substitute the Values

Substitute $k = 3.2$, $b = 3.9$: $f\left(\frac{b}{a}\right) = 3.2 \cdot 2 \cdot 3.9 = 3.2 \cdot 7.8 = 24.96.$

Final Answer

The limit of $y$ as $x \to \frac{b}{a}$ is $24.960$.

Would you like more details on this calculation or have any questions?

Here are some related questions to consider:

1. What is L'Hôpital's Rule, and when can it be used?
2. How do you find the limit of a function as it approaches a specific value?
3. What other methods can be used to resolve indeterminate forms besides L'Hôpital's Rule?
4. Why does direct substitution lead to indeterminate forms?
5. How would the answer change if $k$ had a different value?

Tip: Always check if direct substitution into a limit leads to a determinate value before applying L'Hôpital's Rule or other advanced techniques.