91Ó°ÊÓ

The domain of a function refers to all the possible input values (usually represented by \( x \)) for which the function is defined. In the exercise, we have the function \( f(x) = \frac{1}{\sqrt{x^2-1}} \). The presence of a square root in the denominator introduces restrictions for the domain.

Let's break this down:

• The expression under the square root, \( x^2-1 \), needs to be greater than zero to ensure we get a real number result from the square root. This is because the square root of a negative number is not a real number.
• Hence, the condition \( x^2 - 1 > 0 \) must hold true.
• Solving \( x^2 - 1 = 0 \) gives us the critical points \( x = \pm 1 \), which means the function is undefined at \( x = 1 \) and \( x = -1 \).
• Thus, the domain of the function is \((-\infty, -1) \cup (1, \infty)\).

Identifying the domain is essential because it clarifies where the function can actually be evaluated. In this case, the domain shows that the function cannot take the values \( x = 1 \) and \( x = -1 \), as this would make the denominator zero, leading to undefined results.

Limits are a fundamental concept in calculus used to analyze the behavior of functions as the input approaches certain points. A limit essentially deals with what happens when you get very close to a particular point, not necessarily what happens at that point itself. In this exercise, we look at limits as \( x \) approaches \( 1 \) and \( -1 \).

When checking \( \lim_{x \to 1^+} f(x) \), we see that as \( x \to 1^+ \), \( x^2 - 1 \to 0^+ \), meaning the denominator becomes very small, but positive. This results in \( \frac{1}{\sqrt{x^2 - 1}} \rightarrow \infty \). Thus, the limit is infinite from the right of \( 1 \).

Similar reasoning applies when examining \( \lim_{x \to -1^-} f(x) \). Here, \( x \) is approaching \(-1\) from the left, bringing the denominator again close to zero from the positive side, and thus, the limit becomes infinite as well. This behavior is a characteristic of what we call infinite limits, where the values of the function increase (or decrease) without bound as \( x \) gets close to a certain number. Understanding these limits helps in visualizing the function's behavior near points where it is not defined.

Function analysis involves examining various properties of a function, such as its domain, range, continuity, and behavior near critical points. By analyzing these components, we gain a thorough understanding of how the function behaves across different inputs.

• For the function \( f(x) = \frac{1}{\sqrt{x^2-1}} \), significant analysis focuses on points where the function has infinite limits, namely at \( x = 1 \) and \( x = -1 \). These are points where the function approaches infinity.
• Analysis of such functions often includes looking at one-sided limits to understand the behavior from different directions.
• This further helps to identify whether there's a vertical asymptote, which happens when the function values rise without bound as \( x \) approaches the asymptote from either or both sides.
• By evaluating how the function behaves near these critical points, we develop a clearer understanding of what the graphed function looks like, particularly where it shoots off to infinity, emphasizing its discontinuity at these points.

In conclusion, analyzing functions by focusing on their domains, limits, and behaviors near critical points provides robust insights into their overall characteristics, crucial for solving problems in calculus.