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

Explanation

To evaluate the limit \( \lim_{x \to 0} \frac{3x + |x|}{7x - 5|x|} \), we need to consider the left-hand limit and the right-hand limit separately due to the absolute value function.

### Step 1: Calculate the Left-Hand Limit
The left-hand limit is calculated as \( x \) approaches 0 from the left (i.e., \( x \to 0^- \)). In this case, \( |x| = -x \).

Thus, we have:
\[ \lim_{x \to 0^-} \frac{3x + |x|}{7x - 5|x|} = \lim_{x \to 0^-} \frac{3x - x}{7x + 5x} \]
This simplifies to:
\[ \lim_{x \to 0^-} \frac{2x}{12x} \]
Now, we can simplify further:
\[ \lim_{x \to 0^-} \frac{2}{12} = \frac{1}{6} \]

### Step 2: Calculate the Right-Hand Limit
Next, we calculate the right-hand limit as \( x \) approaches 0 from the right (i.e., \( x \to 0^+ \)). Here, \( |x| = x \).

Thus, we have:
\[ \lim_{x \to 0^+} \frac{3x + |x|}{7x - 5|x|} = \lim_{x \to 0^+} \frac{3x + x}{7x - 5x} \]
This simplifies to:
\[ \lim_{x \to 0^+} \frac{4x}{2x} \]
Now, we can simplify further:
\[ \lim_{x \to 0^+} \frac{4}{2} = 2 \]

### Step 3: Compare the Limits
Now we compare the left-hand limit and the right-hand limit:
- Left-hand limit: \( \frac{1}{6} \)
- Right-hand limit: \( 2 \)

Since the left-hand limit is not equal to the right-hand limit, we conclude that the limit does not exist.

### Final Answer
Thus, the limit \( \lim_{x \to 0} \frac{3x + |x|}{7x - 5|x|} \) does not exist.

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