Let's analyze each of the limits to determine whether the corresponding limit exists as a real number:

1. $\lim_{x \to 81} \frac{x - 81}{\sqrt{x} - 9}$
   This limit has a form that allows for rationalization. Substituting $x = 81$ directly leads to an indeterminate form $\frac{0}{0}$, so we need to manipulate the expression by multiplying the numerator and denominator by the conjugate of the denominator, $\sqrt{x} + 9$.

   After rationalizing and simplifying, this limit exists and is a real number. The checked box is correct.

2. $\lim_{x \to 6} \frac{x^2 + 9x - 90}{x - 6}$
   Substituting $x = 6$ directly results in another indeterminate form $\frac{0}{0}$, so factorization is needed. The quadratic in the numerator can be factored as $(x - 6)(x + 15)$, and the common factor of $x - 6$ cancels out.

   The simplified form allows the limit to be calculated, and it exists as a real number. The checked box is correct.

3. $\lim_{x \to 3} (x + 2) \sin \left( \frac{1}{x - 3} \right)$
   When $x \to 3$, the expression $\sin \left( \frac{1}{x - 3} \right)$ oscillates infinitely, leading to an undefined limit because sine oscillates between -1 and 1, and the factor $\frac{1}{x - 3}$ tends to infinity. Therefore, this limit does not exist as a real number. The unchecked box is correct.

4. $\lim_{x \to 19} \frac{4x - 19}{x - 19}$
   Direct substitution results in $\frac{0}{0}$, an indeterminate form, but the numerator can be simplified as $4(x - 19) + 57$. The common factor of $x - 19$ cancels out, and the limit exists as a real number.

   The checked box is correct.

Conclusion:

The correctly checked options are for limits 1, 2, and 4, where the limits exist as real numbers, while the third limit does not exist.

Would you like further details or have any questions?

Here are some related questions:

1. How do you factor quadratic expressions for limit evaluation?
2. What does it mean for a limit to result in an indeterminate form?
3. How does rationalization help resolve indeterminate forms in limits?
4. What is the behavior of trigonometric limits with oscillatory functions like sine?
5. How can you identify when a limit does not exist?

Tip: Always check for indeterminate forms (like $\frac{0}{0}$) when evaluating limits, and apply algebraic techniques like factoring or rationalization to simplify the expression.