Evaluate `lim_(xto0) (3x+|x|)/(7x-5|x|).`

Explanation

To evaluate the limit \(\lim_{x \to 0} \frac{3x + |x|}{7x - 5|x|}\), we will analyze the left-hand limit and the right-hand limit separately.

### Step 1: Define the limit
We start with the limit expression:
\[ L = \lim_{x \to 0} \frac{3x + |x|}{7x - 5|x|} \]

### Step 2: Calculate the left-hand limit
For the left-hand limit as \(x\) approaches \(0\) (i.e., \(x \to 0^-\)), we substitute \(x = -h\) where \(h\) is a small positive number:
\[ L_{left} = \lim_{h \to 0} \frac{3(-h) + |-h|}{7(-h) - 5|-h|} \]
Since \(|-h| = h\), we can rewrite the expression:
\[ L_{left} = \lim_{h \to 0} \frac{-3h + h}{-7h - 5h} \]
This simplifies to:
\[ L_{left} = \lim_{h \to 0} \frac{-2h}{-12h} = \lim_{h \to 0} \frac{2}{12} = \frac{1}{6} \]

### Step 3: Calculate the right-hand limit
Now, we calculate the right-hand limit as \(x\) approaches \(0\) (i.e., \(x \to 0^+\)), substituting \(x = h\):
\[ L_{right} = \lim_{h \to 0} \frac{3h + |h|}{7h - 5|h|} \]
Since \(|h| = h\), we can rewrite the expression:
\[ L_{right} = \lim_{h \to 0} \frac{3h + h}{7h - 5h} \]
This simplifies to:
\[ L_{right} = \lim_{h \to 0} \frac{4h}{2h} = \lim_{h \to 0} 2 = 2 \]

### Step 4: Compare the limits
Now we compare the left-hand limit and the right-hand limit:
- Left-hand limit: \(L_{left} = \frac{1}{6}\)
- Right-hand limit: \(L_{right} = 2\)

Since \(L_{left} \neq L_{right}\), we conclude that the limit does not exist.

### Final Answer
\[ \lim_{x \to 0} \frac{3x + |x|}{7x - 5|x|} \text{ does not exist.} \]