91Ó°ÊÓ

In calculus, when we want to evaluate limits, the substitution method can be particularly handy. This method is often straightforward as it involves directly substituting the value to which the variable is approaching, into the function.

For instance, in our exercise, we examine what happens when we substitute \(x = -1\) directly into the given rational function \(\frac{x^2+1}{3x^5+4}\). In this step, use basic arithmetic to replace \(x\) with \(-1\) to see if the limit can be trivially evaluated.

In many cases, this direct substitution will lead to an indeterminate form such as \(\frac{0}{0}\) or \(\frac{K}{0}\) (where \(K\) is any non-zero constant), which signals the need for further simplification or different methods. However, if the function simplifies neatly, as it does in this exercise, it allows for a direct evaluation of the limit.

After performing substitution, the next crucial step is simplification. This is the process where the function expression is broken down into its simplest form for easy evaluation.

In our example, after substituting \(x=-1\) into the function, we ended up with \(\frac{2}{-3+4}\), which simplifies straightforwardly to \(\frac{2}{1}\).

Remember:

• Always perform all arithmetic operations, especially addition and subtraction to see if the expression can simplify further.
• Check for any common factors that can be cancelled out in the numerator and the denominator.

In this case, simple arithmetic was performed to arrive at a clear, manageable expression, allowing us to find the limit directly.

Rational functions are those composed of two polynomials, where one polynomial is divided by another. Evaluating their limits is a common exercise in calculus. Understanding the nature of the polynomials involved is key.

For rational functions like \(\frac{x^2+1}{3x^5+4}\), it's important to know what happens as the variable \(x\) approaches a particular value. Typically, if substituting this value into the function does not result in an indeterminate form, the function is continuous at that point, and the value can simply be evaluated.

When dealing with limits:

• Consider the degree of the numerator and the denominator which might impact the behavior of the function at infinity.
• Watch out for cases where both numerator and denominator approach zero, requiring methods like l’Hôpital’s rule for evaluation.

In this exercise, substitution and simplification were sufficient, revealing the limit without further complication.