\(\lim _{\mathrm{x} \rightarrow \infty}\left[\left(\mathrm{x}^{2}+7 \mathrm{x}+2013\right) / \mathrm{x}^{2}\right]^{7 \mathrm{x}}=?\) (a) \(\mathrm{e}^{7}\) (b) \(\mathrm{e}^{14}\) (c) \(\mathrm{e}^{21}\) (d) \(\mathrm{e}^{49}\)

Exponential functions are a fundamental concept in mathematics, often symbolized as functions of the form \( f(x) = a^x \), where 'a' is a constant called the base and 'x' represents the exponent. In problems involving limits, an important exponential function to consider is \( e^x \), where 'e' is Euler's number, approximately equal to 2.71828. This function has a unique property where the rate of change is equal to the value of the function itself.

In the context of limits, particularly in problems like our textbook exercise, an exponent of 'x' or a multiple of 'x' can lead to different behaviors as 'x' approaches infinity. It is crucial to recognize when an expression takes the form of an exponential function during limit evaluation, as it can often simplify the process and point towards utilizing the limit laws or L'Hôpital's rule to find a solution. Understanding the nature of exponential growth—how rapidly something increases as the exponent increases—is helpful in analyzing the behavior of functions as they approach large or small values, often resulting in either infinite limits or limits that approach a specific value based on the properties of 'e'.

For students delving into calculus, L'Hôpital's rule is a reliable tool for tackling indeterminate forms such as 0/0 or \( \infty/\infty \). This rule instructs us to differentiate the numerator and the denominator of a fraction separately and then take the limit again. If the new limit is determinate, or if further application of the rule leads to a determinate limit, then we have solved the original indeterminate form.

However, in our textbook exercise, L'Hôpital's rule does not come into direct play, because we're dealing with an exponent that approaches infinity, not a fraction that leads to an indeterminate form. The rule could be used, though, if through the manipulation of the function, such a scenario arose. As such, understanding when and how to apply L'Hôpital's rule is crucial because it simplifies complex limit problems and offers a pathway to clear-cut answers.

Infinite limits occur when the value of a function grows without bound as the input approaches a certain value. In other words, if as 'x' gets larger and larger (or smaller and smaller), the function's output grows toward positive infinity (\( \infty \)) or drops toward negative infinity (\( -\infty \)), then the limit is said to be infinite.

The textbook exercise provided features an infinite limit since 'x' is approaching infinity. In many cases, the function's behavior can be inferred by evaluating how its components—the constants, variables, and their exponents—behave as 'x' increases. It's important to note that when dealing with polynomial expressions as 'x' approaches infinity, the term with the highest power of 'x' often dominates the behavior of the function.