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ayudame a resolver esto con cada proceso de matematicas correcto:

The problem is asking to analyze the following limit:

\[
\lim_{{x \to 0}} \frac{|x|}{x}
\]

and determine which of the given statements is correct:

a) El límite existe y es 1.  
b) El límite no existe ya que la función no está definida para \(x = 0\).  
c) El límite no existe porque los límites laterales son diferentes.  
d) El límite tiende a infinito.

---

### Step-by-step solution:

1. **Understand the function:**
   \[
   f(x) = \frac{|x|}{x}
   \]
   This is a piecewise function based on the value of \(x\):

   \[
   f(x) =
   \begin{cases} 
   1 & \text{if } x > 0 \\
   -1 & \text{if } x < 0 \\
   \end{cases}
   \]
   
   The function is not defined at \(x = 0\), since division by zero is undefined. Thus, we need to calculate the **left-hand limit** and the **right-hand limit** to see if the overall limit exists.

2. **Right-hand limit (\(x \to 0^+\)):**
   For \(x > 0\), \(|x| = x\), so:
   \[
   \lim_{{x \to 0^+}} \frac{|x|}{x} = \lim_{{x \to 0^+}} \frac{x}{x} = 1
   \]
   Therefore, the right-hand limit is:
   \[
   \lim_{{x \to 0^+}} \frac{|x|}{x} = 1
   \]

3. **Left-hand limit (\(x \to 0^-\)):**
   For \(x < 0\), \(|x| = -x\), so:
   \[
   \lim_{{x \to 0^-}} \frac{|x|}{x} = \lim_{{x \to 0^-}} \frac{-x}{x} = -1
   \]
   Therefore, the left-hand limit is:
   \[
   \lim_{{x \to 0^-}} \frac{|x|}{x} = -1
   \]

4. **Conclusion:**
   Since the right-hand limit is 1 and the left-hand limit is -1, the two limits are not equal. Therefore, the overall limit does not exist.

### Correct Answer:
The correct statement is **c)**: El límite no existe porque los límites laterales son diferentes.

---

Do you want further details or have any questions? Here are some related questions you might explore:

1. What happens if the absolute value function is changed to another type of function?
2. How can you generalize this problem for other types of piecewise functions?
3. What are other cases where limits do not exist due to differing lateral limits?
4. How can this concept be applied in multivariable calculus?
5. Can you give an example where the limit does exist for a piecewise function?

**Tip:** Always remember to check both the right-hand and left-hand limits when determining if a limit exists, especially for piecewise or absolute value functions.

Analiza el siguiente límite: lim(|x|/x) cuando x tiende a 0. ¿Cuál de las siguientes afirmaciones es correcta?

Descubre cómo analizar el límite lim(|x|/x) cuando x tiende a 0 y determinar cuál es la afirmación correcta sobre la existencia del límite. Este problema aborda conceptos clave en cálculo, incluyendo límites y funciones por partes. Apto para estudiantes de los grados 11-12.

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