91Ó°ÊÓ

In calculus, one-sided limits help us determine the behavior of a function as it approaches a specific point from only one direction—either from the left or right. For the function \( \frac{1}{x^2 - 4} \), this means observing what happens as \( x \) nears certain critical values from one side.

In our exercise, we computed the limits as \( x \) approaches 2 and -2 from both the right (positive side) and the left (negative side). Here’s how it works:

• For \( x \to 2^+ \), the expression is analyzed as \( x \) gets close to 2 from the right-hand side, affecting the sign of the factor \( x-2 \).
• For \( x \to 2^- \), it focuses on approaching 2 from the left-hand side. Again, it's the sign of \( x-2 \) that's key here.
• The same concept applies when we look at \( x \to -2^+ \) and \( x \to -2^- \), determining how the factors in the denominator behave as \( x \) comes close to -2 from the right and left respectively.

This directional approach to limits is useful for understanding asymptotic behavior and discontinuities.

When a function grows larger and larger without bound as it approaches a certain value, we say it has an infinite limit.

In our exercise concerning \( \frac{1}{x^2 - 4} \):

• As \( x \to 2^+ \), the denominator \((x-2)(x+2)\) approaches zero from the positive side, leading to the function tending towards \(+\infty\) because dividing by a very small positive number results in a very large positive number.
• On the other hand, when \( x \to 2^- \), the same denominator approaches zero from the negative side, so the limit becomes \(-\infty\) as division by a very small negative number results in a very large negative number.
• Similarly, as \( x \to -2^+ \), the denominator heads toward zero from the negative side, making the limit \(-\infty\).
• Finally, when \( x \to -2^- \), the denominator approaches zero from the positive side, causing the limit to be \(+\infty\).

Through infinite limits, we can see how functions behave as their values increase or decrease without bound, often resulting in vertical asymptotes.

Factoring is a mathematical tool that can greatly simplify the process of evaluating limits, especially when the function involves polynomial expressions.

For example, the function in our exercise is \( \frac{1}{x^2 - 4} \). By factoring the denominator, we have:\[ x^2 - 4 = (x-2)(x+2) \]This factorization converts the function into a product, making it easier to analyze how the expression behaves as \( x \) approaches critical values like 2 and -2.

• It helps us clearly identify points where the denominator becomes zero, which are potential points of discontinuity or undefined behavior.
• Understanding the factorization allows us to determine the direction from which the denominator approaches zero, which is crucial for correctly evaluating one-sided limits and determining whether they lead to positive or negative infinity.

By using factoring, we achieve a clearer picture of the function's behavior near points of interest, facilitating the computation of limits. This is especially helpful when dealing with rational functions and helps avoid computational errors.