Question

Finding a Limit In Exercises \(7-26\) , find the limit (if it exists). If it does not exist, explain why. $$ \lim _{x \rightarrow-3^{-}} \frac{x}{\sqrt{x^{2}-9}} $$

Short answer

The limit as \( x \) approaches \( -3 \) from the left of the function \( \frac{x}{\sqrt{x^{2}-9}} \) is \( -\infty \), and is not defined for \( x \) approaching \( -3 \) overall, since it approaches \( +\infty \) from the right side.

Step-by-step solution

Arrange the Function

In order to simplify the limit analysis, we can rewrite our function as \( \frac{x}{\sqrt{x^{2}-9}} = \frac{x}{\sqrt{(x + 3)(x - 3)}} \) taking the difference of two squares into account

Compute Limit

We now find the limit as \( x \) approaches \( -3 \) from the left: \( \lim _{x \rightarrow-3^{-}} \frac{x}{\sqrt{(x + 3)(x - 3)}} \) . Our denominator \( \sqrt{ (x + 3)(x - 3) } \) will approach 0 because \( x + 3 \) will approach zero from the left as \( x \) approaches \( -3 \) , so overall \( \lim _{x\rightarrow -3^{-}}(x+3)(x-3) \) will approach 0 from the negative side, since \( (x-3) < 0 \) when \( x < 3 \). The top part will also approach \( -3 \), but we are not dividing by zero since we are approaching from the left-side of \( -3 \), not exactly approaching to \( -3 \). Provided that the numerator does not go to zero as fast as the denominator, the limit is \(-\infty\)

Check Other Direction

Note that approaching \( -3 \) from the other side would give a different result. As we approach \( -3 \) from the right side, the value \( x -3 \) is still negative (as \( x \) is less than \( 3 \)), but \( x + 3 \) is now positive (as \( x \) is more than \( -3 \)), which would mean an overall positive root, so the whole fraction now goes to positively infinity, instead of negatively. Thus, the limit as \( x \) approaches \( -3 \) is not defined.

Key concepts

One-sided limits

When we talk about one-sided limits, we refer to the value a function approaches as the input approaches a particular point from one specific direction — either from the left or the right. In our problem, we're interested in what happens as \( x \) approaches \(-3\) from the left side, denoted as \( x \rightarrow -3^{-} \). This means we're considering values of \( x \) that are slightly less than \(-3\).

Understanding one-sided limits helps us tackle functions that aren't well-behaved or continuous at certain points. By analyzing them from one direction at a time, we can more easily determine the behavior of the function near that point.

• For the given function, \( \frac{x}{\sqrt{(x + 3)(x - 3)}} \), approaching \(-3\) from the left means \( x + 3 \) is negative.
• The numerator \( x \) approaches \(-3\), while the significant change happens in the denominator, where \( \sqrt{(x + 3)(x - 3)} \) approaches zero.
• Because the approach is only from one side, different behaviors can emerge from different directions, leading to potentially undefined or infinite limits.

Indeterminate forms

An indeterminate form occurs when evaluating a limit results in a mathematical expression that doesn’t initially give a clear-cut value. Common indeterminate forms include \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). These forms often require additional steps to simplify or rearrange the function to find the limit.

In this exercise, as \( x \) approaches \(-3\), our expression is initially of the form \( \frac{-3}{0} \). While this isn't one of the classic indeterminate forms, it still requires careful handling because dividing by zero is undefined. Rather than a simple result, the behavior and limit require an investigation of the function’s behavior near the point.

• The limit analysis must consider how both the numerator and denominator behave as \( x \) approaches \(-3\).
• Indeterminate forms often signal that the function needs to be rewritten or analyzed in parts, as was done using the difference of squares \((x + 3)(x - 3)\).
• Such forms challenge us to think more deeply about continuity and discontinuity in functions.

Limits approaching infinity

Limits approaching infinity help us understand how a function behaves when the input grows very large or very small. Here, the concept is useful even when the input itself isn’t approaching infinity, but when the function's value does. This occurs in our solution, as the limit of the function as \( x \to -3^{-} \) results in \(-\infty\).

Finding the limit involves deeply examining the rates of change in both the numerator and the denominator.

• In this case, even though \( x \) only nears \(-3\), the changes result in the expression shooting toward infinity negatively.
• This infinite behavior occurs because the denominator approaches zero faster than the numerator.
• Understanding these principles gives us a fuller picture of limits and prepares us for calculus concepts, like asymptotic behaviors and horizontal asymptotes.
• Analyzing how and why the function does or does not reach certain values sharpens our computational skills and theoretical insight.