Question

Find the limit. \(\lim _{x \rightarrow \infty}\left(\frac{2 x^{2}}{x-1}+\frac{3 x}{x+1}\right)\)

Short answer

The limit of the function as x approaches infinity is \(\infty\).

Step-by-step solution

Simplify the Function

The function \(\lim _{x \rightarrow \infty}\left(\frac{2 x^{2}}{x-1}+\frac{3 x}{x+1}\right)\) can be simplified by dividing each term by \(x^{2}\). This is particularly useful when dealing with polynomial functions. You'll obtain: \(\lim _{x \rightarrow \infty}\left(\frac{2 x^{2}}{x^{2}-x}+\frac{3 x}{x^{2}+x}\right)\). Then further simplify it to: \(\lim _{x \rightarrow \infty}\left(\frac{2}{1-\frac{1}{x}}+\frac{3}{\frac{1}{x}+\frac{1}{x^{2}}}\right)\).

Find the values

As x approaches infinity, \( \frac{1}{x} \) and \( \frac{1}{x^{2}} \) approaches 0. So, substitute these values into the simplified function. You'll then get: \(\frac{2}{1-0}+\frac{3}{0+0}\)

Final answer

This simplifies to \(2 + \infty\) or simply \(\infty\). The limit of the function as x approaches infinity does not exist or it equals \(\infty\) in this case.

Key concepts

Limit at Infinity

Understanding the concept of the 'limit at infinity' is an essential part of calculus. It helps us grasp the behavior of functions as the inputs grow without bound. In simple terms, when we talk about the limit at infinity, we're asking what value a function approaches as the variable, usually denoted by 'x', heads towards positive or negative infinity.

For example, in the given exercise ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ``, the function presented combines two rational expressions where x increases without limit. The approach to evaluate this is essentially looking at the highest powers of x in the numerator and denominator, as they are the ones that dictate the function's behavior at extreme values of x. If the highest power in the numerator is greater than that in the denominator, the limit will typically be infinity, reflecting the dominance of the numerator's growth. In our case, the degrees of the polynomial in the numerator and denominator give us a clear indication: the limit is infinite.

Polynomial Functions

Polynomial functions are algebraic expressions that involve sums of powers of a variable. These functions are particularly easy to work with when finding limits at infinity because their behavior is dominated by the term with the highest power.

In the context of our exercise, the function is composed of two separate polynomial expressions within the structure of rational functions. When finding the limit as x approaches infinity for polynomials, the term with the highest exponent on x is the most significant, and other terms become negligible. For instance, in a function like ewline$$ ewline ewline$$, what really matters when x gets very large is the highest power of x, which is ewline$$ in the first term and ewline$$ in the second term. The strategy of dividing each term by the highest power present is used to make this observation more evident and simplifies the process of evaluating the limit at infinity.

Simplifying Expressions

Simplifying expressions is a crucial skill in calculus, especially when dealing with complex functions. The simpler a function, the easier it is to manipulate and understand its traits, including finding its limits.

As we carry out the simplification in our exercise, dividing by the highest power of x serves to normalize the expressions, breaking them down into more manageable pieces. This technique is beneficial when the function's terms all contain powers of x, hence why we adapt the expression by introducing factors of 1/x. After simplification, which is step one in our solution, values that seem complicated are reduced to more straightforward expressions like ewline$$ or ewline$$, vastly simplifying the task of evaluating the limit. In calculus, such simplification not only makes it easier to solve problems but also to understand the underlying principles guiding a function's behavior.

Indeterminate Forms

In calculus, certain expressions are not immediately evident in defining their limit, referred to as 'indeterminate forms'. Common examples include 0/0, ewline$$, and ewline$$, among others. These forms can occur in more complex limits scenarios and require additional algebraic manipulation or other methods such as L'Hôpital's rule to resolve.

However, in the given problem, we encounter a less common indeterminate form as we reach step two of our solution: ewline$$, which simplifies down to ewline$$ plus ewline$$, implicating an undefined situation. This term points out that the function grows beyond any bound as x approaches infinity. While some may see this as an indeterminate form, in the context of this exercise, it represents the concept of approaching infinite behavior rather than an expression that needs further evaluation to find a finite limit.