To solve the given limit, we start by defining the limit expression:

$\lim_{x \to 1} \frac{\sqrt{x+2} - 4}{5x^2 + x - 2}$

Step 1: Evaluate the expression at $x = 1$

Substituting $x = 1$ into both the numerator and the denominator:

Numerator: $\sqrt{1 + 2} - 4 = \sqrt{3} - 4$

Denominator: $5(1)^2 + 1 - 2 = 5 + 1 - 2 = 4$

Step 2: Check for indeterminate form

The numerator evaluates to $\sqrt{3} - 4$, which is a finite number. The denominator evaluates to 4, which is also a finite number. Since there is no indeterminate form (e.g., $\frac{0}{0}$ or $\frac{\infty}{\infty}$), we can substitute $x = 1$ directly into the limit.

Step 3: Substitute $x = 1$ directly into the limit

$\lim_{x \to 1} \frac{\sqrt{x+2} - 4}{5x^2 + x - 2} = \frac{\sqrt{3} - 4}{4}$

Final Answer

$\lim_{x \to 1} \frac{\sqrt{x+2} - 4}{5x^2 + x - 2} = \frac{\sqrt{3} - 4}{4}$

This fraction is the exact answer to the limit expression.

Would you like more details on any step or have any questions?

Here are some related questions:

1. How do you handle limits that result in indeterminate forms?
2. What are the common techniques for solving limits involving square roots?
3. How does L'Hôpital's Rule help solve limits with indeterminate forms?
4. What is the difference between evaluating a limit directly and using algebraic manipulation?
5. How can limits be used to find the derivatives of functions?

Tip: When evaluating limits, always check first whether direct substitution results in an indeterminate form. If it doesn't, direct substitution can often give you the answer quickly.