lim of x approfhes to negative 0 from 1/x sqare ori e  at 1/x and then for f(x)=m find the real values of m for each the equations has exactly 3 real solutions

Let's break this problem into two parts and solve them step-by-step.

### Part 1: Finding the Limit

First, we need to find the limit of \( \frac{1}{x^2} \) as \( x \) approaches \( 0 \) from the negative side.

\[ \lim_{{x \to 0^-}} \frac{1}{x^2} \]

The function \( \frac{1}{x^2} \) is defined as \( x \) approaches \( 0 \) from the negative side. Notice that \( x^2 \) is always positive regardless of the sign of \( x \). As \( x \) approaches \( 0 \) from the negative side, \( x^2 \) approaches \( 0 \) and hence \( \frac{1}{x^2} \) becomes infinitely large. Therefore, the limit is:

\[ \lim_{{x \to 0^-}} \frac{1}{x^2} = \infty \]

### Part 2: Finding \( m \) for Exactly 3 Real Solutions

Next, we need to determine the real values of \( m \) for which the equation \( f(x) = m \) has exactly three real solutions, given the function \( f(x) \).

Assume \( f(x) = \frac{1}{x} \). We need to find the real values of \( m \) for which the equation \( \frac{1}{x} = m \) has exactly three real solutions.

The equation \( \frac{1}{x} = m \) simplifies to:

\[ x = \frac{1}{m} \]

For this equation to have exactly three real solutions, we need a bit more information or a different form of the function \( f(x) \). However, based on the standard interpretation of the problem and usual functions, let's assume \( f(x) = x^3 \). In this case, the equation becomes:

\[ x^3 = m \]

The cubic function \( x^3 = m \) will intersect a horizontal line \( y = m \) exactly three times if and only if \( m = 0 \) (which means the line touches the origin where the function has an inflection point). Otherwise, for a typical cubic polynomial, the intersections depend on the specific form of the function. However, given the typical cubic polynomial scenario, \( m = 0 \) is the special case that we need. Hence, there is only one \( m \):

\[ m = 0 \]

### Summary

1. The limit as \( x \) approaches \( 0 \) from the negative side of \( \frac{1}{x^2} \) is \( \infty \).
2. For the function \( f(x) = x^3 \), the value of \( m \) for which the equation \( f(x) = m \) has exactly three real solutions is \( m = 0 \).

Would you like more details or have any questions about these solutions? Here are 5 additional questions you might consider:

1. How do we find the limits of more complex functions?
2. What are the properties of cubic functions and their roots?
3. How do you solve equations involving higher-degree polynomials?
4. Can you explain how the Intermediate Value Theorem helps in finding roots?
5. What is the significance of inflection points in polynomial functions?

**Tip:** When dealing with limits involving infinity, always consider the behavior of the function as it approaches the point from both sides to understand the full behavior of the function.

Explore the limit of 1/x^2 as x approaches 0 and solving cubic functions to find real values of m with exactly three solutions. This problem involves key concepts in limits, cubic functions, and real solutions in mathematics.

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