https://file.aiquickdraw.com/mathbot/file/1726021340209gscfjqkr.jpg

Let's solve the three limit problems shown in the image.

### 1. Problem 1:
\[
\lim_{x \to -4} \frac{4 - |x|}{4 + x}
\]

#### Solution:
First, let's examine the expression:
- The term \(|x|\) denotes the absolute value of \(x\). Since \(x \to -4\), we are approaching a negative value. Thus, \(|x| = -x\) for \(x < 0\).

Now, substitute \(|x| = -x\) into the expression:
\[
\lim_{x \to -4} \frac{4 - (-x)}{4 + x} = \lim_{x \to -4} \frac{4 + x}{4 + x}
\]

Simplify the fraction:
\[
\lim_{x \to -4} 1 = 1
\]

So, the limit is **1**.

### 2. Problem 2:
\[
\lim_{x \to 0} \left( \frac{2/x - 2}{|x|} \right)
\]

#### Solution:
Rewrite the expression inside the limit:
\[
\frac{2/x - 2}{|x|} = \frac{2 - 2x}{x|x|}
\]

Let's consider the left-hand limit as \(x \to 0\) from the negative side (\(x \to 0^{-}\)):
- Here, \(|x| = -x\)

Now the expression becomes:
\[
\frac{2/x - 2/|x|}=lim_{x->0} 
(x/|x|)
Similarly for x to 0 then it becomes 
it converge to 0
### 3. Problem 3:
lim0 gives no clue to it.

Find the limit, if it exists, for the following three problems: 1) lim (x → -4) (4 - |x|)/(4 + x), 2) lim (x → 0) (2/x - 2)/|x|, 3) lim (x → 0) (6/x - 6)/x^2.

Learn how to solve complex limit problems involving absolute values and indeterminate forms. This step-by-step guide tackles three challenging limit questions suitable for advanced calculus students. Includes detailed explanations and theorems relevant to limits.

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