If \(\lim _{\mathrm{x} \rightarrow \infty}\left[\left\\{\mathrm{x}^{2}(\mathrm{~m}-1)-(\mathrm{m}+\mathrm{n}) \mathrm{x}-2013\right\\} /(\mathrm{x}+1)\right]=1\) then values of constant \(\mathrm{m}=\ldots \ldots\) and \(\mathrm{n}=\ldots \ldots\) (a) \(\mathrm{m}=1, \mathrm{n}=0\) (b) \(\mathrm{m}=1, \mathrm{n}=-2\) (c) \(\mathrm{m}=1, \mathrm{n}=1\) (d) \(\mathrm{m}=1, \mathrm{n}=-1\)

In calculus, indeterminate forms arise when evaluating limits that can potentially take on many different values. They occur in the context of finding the limit of a function at points where the function itself does not provide a clear answer.

For example, the expression \(\frac{0}{0}\) is considered an indeterminate form. It's not simply zero because we could be dealing with a situation where both the numerator and the denominator are approaching zero at different rates. Other indeterminate forms include \(\frac{\infty}{\infty}\), \(0 \cdot \infty\), \(\infty - \infty\), \(1^\infty\), \(0^0\), and \(\infty^0\).

Techniques like factoring, simplification, or more advanced methods, such as L'Hopital's Rule, are often employed to resolve these indeterminate forms and find the true limit of a function.

Limits to infinity involve evaluating the behavior of a function as the independent variable heads towards positive or negative infinity. These limits are fundamental in understanding the end behavior of a function and are essential for analyzing and interpreting the long-term trends of mathematical models.

When a limit is taken as \(x\) approaches infinity, it intuitively looks at the value the function approaches as \(x\) grows without bound. In the given problem, evaluating \(\lim _{x \rightarrow \infty}\) involves dealing with a polynomial of the form \(x^2(m-1)-mx-2013\). This leads to determining the leading coefficient and degree of the polynomial, to understand if the function grows without bound or approaches a finite value, which in this case, is 1.

L'Hopital's Rule is a powerful tool in calculus used to find limits of indeterminate forms such as \(\frac{0}{0}\) and \(\frac{\infty}{\infty}\). If a limit yields an indeterminate form when directly substituted, L'Hopital's Rule states that you can differentiate the numerator and the denominator separately and then retry taking the limit.

For the limit to be valid after applying L'Hopital’s Rule, it is required that both derivatives exist and the limit of their quotient must exist or be \(\pm \infty\). It's imperative to continue applying L'Hopital's Rule until a determinate form is obtained. This rule can often simplify complex limits, but it should be used only when necessary, as there might be simpler methods to evaluate a limit.