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Hyperbolic functions are analogs to the traditional trigonometric functions but are based on the hyperbola instead of the circle. The two fundamental hyperbolic functions are:

• The hyperbolic sine, noted as \( \sinh(x) = \frac{e^{x} - e^{-x}}{2} \).
• The hyperbolic cosine, noted as \( \cosh(x) = \frac{e^{x} + e^{-x}}{2} \).

The hyperbolic tangent, \( \tanh(x) \), combines these two functions as a ratio: \( \tanh(x) = \frac{\sinh(x)}{\cosh(x)} \). These functions share properties similar to trigonometric functions, such as derivatives and integrals, but instead of modeling oscillations like sine and cosine, they relate to exponential growth and decay. In calculus, hyperbolic functions help describe certain kinds of curves in physics and engineering, making them very useful in various applications.

Exponential functions are functions of the form \( f(x) = a^x \), where \( a \) is a constant and \( x \) is the variable. A special case is when \( a = e \), the base of natural logarithms, approximated by 2.718. This function is crucial due to its characteristic property: the rate of growth is proportional to the value of the function. This makes it a fundamental model in fields such as population growth, radioactive decay, and finance.
\( e^x \) tends to infinity as \( x \) increases, and \( e^{-x} \) tends to zero. This property creates a balance for expressions like \( e^{x} - e^{-x} \) and \( e^{x} + e^{-x} \) when considering limits at infinity.
Understanding these differences is key for simplifying expressions involving hyperbolic functions as they can change rapidly as \( x \) becomes very large or small. In turn, this impacts the behavior of related limits.

In calculus, the concept of infinity is used to describe a behavior where a variable grows without bound. It doesn't represent a number but rather an idea about growth and limits. Consider limits where \( x \) approaches infinity. Functions can either increase or decrease boundlessly or approach a constant value.
When we say \( \lim_{x \to \infty} \frac{1}{x^3} = 0 \), it describes how as \( x \) gets larger, \( \frac{1}{x^3} \) becomes smaller and smaller, approaching zero but never actually reaching it. This is because dividing 1 by an ever-increasing number approaches zero in value.
In expressions involving infinity, it helps to break down a function into simpler parts—such as isolating exponential components—enabling us to see how each term behaves on its own before combining them back to see the overall effect. Recognizing these behaviors is essential as it allows us to predict and calculate the limits accurately.