The statement "if $f(x) > 0$ for all real numbers, then the limit of $f(x)$ as $x \to c$ is greater than 0" is not true. To disprove this, it suffices to find a counterexample.

Counterexample:

Consider the function:

$f(x) = \frac{1}{x^2 + 1}$

1. For all real numbers $x$, we can observe that $f(x) > 0$. Specifically:

$f(x) = \frac{1}{x^2 + 1} > 0$

since $x^2 + 1$ is always greater than 0 for any real $x$.

2. Now, let’s compute the limit of $f(x)$ as $x \to \infty$:

$\lim_{x \to \infty} f(x) = \lim_{x \to \infty} \frac{1}{x^2 + 1} = 0$

Conclusion:

Even though $f(x) > 0$ for all real $x$, the limit of $f(x)$ as $x \to \infty$ is 0, which disproves the statement.

Summary:

The claim that "if $f(x) > 0$ for all $x$, then $\lim_{x \to c} f(x) > 0$" is false because a function can be positive for all $x$ but still approach 0 in the limit.

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Do you want further details or have any questions on this?

Here are some related questions:

1. What conditions ensure that the limit of a function remains positive?
2. Can a function approach a negative limit if it's always positive?
3. How does the behavior of rational functions influence their limits?
4. What are other examples of functions that are positive but tend to zero?
5. How can limits involving infinity be evaluated?

Tip: When proving something false, finding a single counterexample is enough.