The mathematical expression shown in the image is the limit:

$\lim_{x \to \infty} \frac{x^2 - 4x + 3}{\sqrt{2x + 7} - \sqrt{x + 8}}$

Solution:

We aim to evaluate the above limit.

Step 1: Simplify the denominator

To simplify, multiply the numerator and denominator by the conjugate of the denominator:

$\sqrt{2x + 7} - \sqrt{x + 8} \text{ (original denominator) becomes } \sqrt{2x + 7} + \sqrt{x + 8}.$

This gives: $\frac{x^2 - 4x + 3}{\sqrt{2x + 7} - \sqrt{x + 8}} \cdot \frac{\sqrt{2x + 7} + \sqrt{x + 8}}{\sqrt{2x + 7} + \sqrt{x + 8}}.$

The denominator becomes: $(\sqrt{2x + 7})^2 - (\sqrt{x + 8})^2 = (2x + 7) - (x + 8) = x - 1.$

So the expression simplifies to: $\frac{(x^2 - 4x + 3)(\sqrt{2x + 7} + \sqrt{x + 8})}{x - 1}.$

Step 2: Analyze the dominant terms as $x \to \infty$

• For the numerator $x^2 - 4x + 3$, the leading term is $x^2$.
• For the denominator $x - 1$, the leading term is $x$.

Thus, the fraction simplifies to: $(x^2 - 4x + 3)(\sqrt{2x + 7} + \sqrt{x + 8}) \sim x^2 \cdot (\sqrt{2x} + \sqrt{x}) = x^2 (\sqrt{x}( \sqrt{2} + 1)).$

The leading terms dominate.

Step 3: Conclude

Finally expand clearer pathsLet’s proceed systematically with the dominant term analysis. Our expression after simplifying the denominator becomes:

$\frac{(x^2 - 4x + 3)(\sqrt{2x + 7} + \sqrt{x + 8})}{x - 1}.$

Step 1: Factorize dominant terms in the numerator

For large $x$:

• $x^2 - 4x + 3 \approx x^2$,
• $\sqrt{2x + 7} \approx \sqrt{2x}$,
• $\sqrt{x + 8} \approx \sqrt{x}$.

So, the numerator is approximately: $(x^2)(\sqrt{2x} + \sqrt{x}) = x^2 \cdot \sqrt{x} (\sqrt{2} + 1).$

Step 2: Simplify the denominator

For large $x$, the denominator $x - 1 \approx x$. Thus, the fraction becomes: $\frac{x^2 \cdot \sqrt{x} (\sqrt{2} + 1)}{x}.$

Step 3: Simplify further

Divide $x^2 \cdot \sqrt{x}$ by $x$: $\frac{x^2 \cdot \sqrt{x}}{x} = x \cdot \sqrt{x} = x^{3/2}.$

Therefore, the limit becomes: $\lim_{x \to \infty} x^{3/2} (\sqrt{2} + 1).$

Conclusion

The limit diverges to $+\infty$, as $x^{3/2}$ grows unbounded. Hence, the result is:

$\text{Limit } = +\infty.$

Would you like me to break this into more detailed steps or explore any related limits?

Follow-up Questions

1. How does multiplying by the conjugate help in simplifying expressions with square roots?
2. What are some common dominant term approximations for $x \to \infty$?
3. How does the behavior of $x^{3/2}$ compare to $x^n$ for other powers $n$?
4. Can we apply L’Hôpital’s Rule to verify the divergence of the limit?
5. How would the solution change if the denominator were slightly different, e.g., $\sqrt{2x + 7} - \sqrt{4x + 8}$?

Tip

When dealing with limits involving square roots, rationalizing the denominator (or numerator) by multiplying by the conjugate is a standard technique to simplify the expression.