https://file.aiquickdraw.com/mathbot/file/17249250643526b2zwinu.jpg

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.

Explore detailed solutions to limit problems and expression evaluations in calculus. Learn about trigonometric identities, factoring, and L'Hôpital's rule through practical examples. Suitable for college-level students studying calculus.

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