91Ó°ÊÓ

The substitution method is a handy technique in calculus for finding the limit of a function. The idea is simple: replace the variable with the given value and then simplify the expression. It's a straightforward way to see if the function approaches a finite value.
For instance, when you have \(lim _{x \rightarrow c} f(x)\), you can substitute \(x = c\) and evaluate the expression directly. If the resulting expression gives a valid number, hooray! That's your limit.
However, sometimes direct substitution doesn't work, especially when it leads to an indeterminate form like \(\frac{0}{0}\). In such cases, more work may be needed, including simplification or other methods like L'Hopital's Rule.

A rational function is simply a fraction where both the numerator and the denominator are polynomials. They have a lot of interesting characteristics, especially when it comes to limits.
Depending on the degrees of the polynomials, the behavior of rational functions can change significantly.

• If both polynomials have the same degree, the limit is usually the ratio of the leading coefficients.
• If the numerator has a higher degree, the limit tends to infinity.
• And if the denominator's degree is higher, the limit approaches zero.

For the given exercise, your aim was to find the limit of a rational function as \(x\) approaches \( -\frac{1}{2}\). Rational functions often involve simplifying or factoring to make evaluation easier, but sometimes direct substitution works without any extra steps.

Evaluating limits tells us how a function behaves as it nears a specific input value. It's all about understanding the function's behavior, not just at \(x\), but around \(x\) as well.
In many cases, you can often directly substitute the value into the function. This works beautifully if it doesn’t make the expression undefined. However, if it does, that's when you'll need other strategies.
In this example, you directly evaluated the function's limit by substituting \(x = -\frac{1}{2}\). The calculation gave a straightforward result of \(2\), thanks to the function's smooth behavior at that point. It's important to know that limits involve investigating behavior from both sides of \(x\), even if sometimes you need only focus on a one-sided approach, like here.