91Ó°ÊÓ

Understanding dominant terms is crucial in calculus, especially when dealing with limits involving infinity. Dominant terms are those with the highest powers of a variable, typically x in our case. They have the greatest impact on the value of the function as the variable grows larger or smaller.
In the exercise, we identified that the dominant term in both the numerator and the denominator is \(x^2\). This is because \(x^2\) grows faster than any lower power of x, such as \(x\) or a constant like 1.

By focusing on these dominant terms, we can often simplify complex expressions, making it easier to compute limits. Always look for the terms that will 'dominate' the behavior of the function as the variable approaches large values, like infinity.

Simplifying expressions before evaluating limits can make the process much easier. In the given limit \(\lim _{x \rightarrow \infty} \frac{x^{2}+x}{x^{2}-1}\), the key step was to simplify the fraction.

We do this by dividing both the numerator and the denominator by \(x^2\), the dominant term. This gives:
\[ \frac{x^2 + x}{x^2 - 1} = \frac{x^2/x^2 + x/x^2}{x^2/x^2 - 1/x^2} = \frac{1 + \frac{1}{x}}{1 - \frac{1}{x^2}} \]
This simplification helps because it reduces the expression to a form where the terms involving \(\frac{1}{x}\) and \(\frac{1}{x^2}\) can be more easily handled as x approaches infinity.
Notice how much cleaner and simpler the expression becomes, making the next steps more straightforward.

Evaluating limits involves understanding the behavior of the function as the variable approaches a certain value. In this specific exercise, we were dealing with \(\lim_{x \rightarrow \infty}\).
After simplifying the expression to \(\frac{1 + \frac{1}{x}}{1 - \frac{1}{x^2}}\), we consider how \(\frac{1}{x}\) and \(\frac{1}{x^2}\) behave as \(x \rightarrow \infty\):

• \(\frac{1}{x} \rightarrow 0\)
• \(\frac{1}{x^2} \rightarrow 0\)

Substituting these limits into our simplified expression, we get:
\[ \frac{1 + 0}{1 - 0} = 1 \]
Therefore, the limit is 1.
Evaluating limits often involves recognizing the behavior of functions and expressions as the variable approaches a particular value. Understanding this process deeply enhances your problem-solving skills in calculus.