Let's solve each problem from the image one by one.

1. Evaluate $\lim_{x \to 0} \left( 1 - \sin^2(\theta) \right)^{1/2}$

This expression can be simplified using a trigonometric identity.

$1 - \sin^2(\theta) = \cos^2(\theta)$

So, the expression becomes:

$\lim_{x \to 0} \sqrt{\cos^2(\theta)} = \lim_{x \to 0} |\cos(\theta)|$

As $\theta$ approaches 0, $\cos(0) = 1$. Therefore, the limit is:

$\boxed{1}$

2. Simplify the expression $\lim_{x \to 4} \frac{x^2 - 16}{x - 4}$

This expression involves a limit that can be simplified by factoring the numerator.

Factor the numerator $x^2 - 16$ as:

$x^2 - 16 = (x - 4)(x + 4)$

Thus, the expression becomes:

$\lim_{x \to 4} \frac{(x - 4)(x + 4)}{x - 4}$

Cancel out the common factor $x - 4$:

$\lim_{x \to 4} (x + 4)$

Now, substitute $x = 4$:

$4 + 4 = \boxed{8}$

3. Evaluate the following limit $\lim_{x \to 1} \frac{x^2 - 1}{x^2 + 3x - 4}$

First, factor both the numerator and the denominator:

• Numerator: $x^2 - 1 = (x - 1)(x + 1)$
• Denominator: $x^2 + 3x - 4 = (x - 1)(x + 4)$

The expression becomes:

$\lim_{x \to 1} \frac{(x - 1)(x + 1)}{(x - 1)(x + 4)}$

Cancel out the common factor $x - 1$:

$\lim_{x \to 1} \frac{x + 1}{x + 4}$

Substitute $x = 1$:

$\frac{1 + 1}{1 + 4} = \frac{2}{5} = \boxed{\frac{2}{5}}$

4. Evaluate the limit \lim_{x \to 4} \frac{(x - 4)/(x^2 - x - 12)}

First, factor the quadratic expression in the denominator:

• Denominator: $x^2 - x - 12 = (x - 4)(x + 3)$

The expression becomes:

$\lim_{x \to 4} \frac{x - 4}{(x - 4)(x + 3)}$

Cancel out the common factor $x - 4$:

$\lim_{x \to 4} \frac{1}{x + 3}$

Substitute $x = 4$:

$\frac{1}{4 + 3} = \frac{1}{7} = \boxed{\frac{1}{7}}$

5. Evaluate the limit $\lim_{x \to \infty} \frac{\ln x}{x}$

For this limit, as $x \to \infty$, $\ln x$ grows much slower than $x$. Thus, the limit is:

$\lim_{x \to \infty} \frac{\ln x}{x} = 0 = \boxed{0}$

Would you like to continue with the remaining problems or have any other questions?

Here are 5 related questions:

1. How would the limit in problem 1 change if $\theta$ approaches $\frac{\pi}{2}$?
2. What happens to the expression in problem 2 if we change the limit to $x \to -4$?
3. How can L'Hôpital's rule be applied to problem 3?
4. What does the expression in problem 4 evaluate to if the quadratic expression was not factorable?
5. What would happen to the limit in problem 5 if we considered $\lim_{x \to 0^+}$?

Tip: Factoring and simplifying before taking limits often helps avoid indeterminate forms and simplifies the evaluation process.