For $f(x)$ Given Below, Evaluate Lim ⁡ X → ∞ F ( X \lim _{x \rightarrow \infty} F(x Lim X → ∞ ​ F ( X ] And Lim ⁡ X → − ∞ F ( X \lim _{x \rightarrow-\infty} F(x Lim X → − ∞ ​ F ( X ]. F ( X ) = − 4 + X 4 − 2 X 3 − 5 X + X 2 + 1 F(x)=\frac{-4+x^4-2 X^3}{-5 X+x^2+1} F ( X ) = − 5 X + X 2 + 1 − 4 + X 4 − 2 X 3 ​

Introduction

In calculus, limits are used to study the behavior of functions as the input values approach a specific point. In this article, we will evaluate the limits of a given rational function as x approaches positive and negative infinity. The function is given by:

$$f(x)=\frac{-4+x^4-2 x^3}{-5 x+x^2+1}$$

Limit as x Approaches Positive Infinity

To evaluate the limit as x approaches positive infinity, we need to analyze the behavior of the function as x becomes very large. We can start by dividing both the numerator and denominator by the highest power of x, which is $x^4$.

\lim _{x \rightarrow \infty} f(x) = \lim _{x \rightarrow \infty} \frac{\frac{-4}{x^4}+1-\frac{2}{x}}{-\frac{5}{x^3}+1+\frac{1}{x^2}}

As x approaches positive infinity, the terms $\frac{-4}{x^4}$, $\frac{2}{x}$, $\frac{5}{x^3}$, and $\frac{1}{x^2}$ all approach 0. Therefore, the limit simplifies to:

\lim _{x \rightarrow \infty} f(x) = \lim _{x \rightarrow \infty} \frac{1}{1} = 1

Limit as x Approaches Negative Infinity

To evaluate the limit as x approaches negative infinity, we can use a similar approach. We divide both the numerator and denominator by the highest power of x, which is $x^4$.

\lim _{x \rightarrow -\infty} f(x) = \lim _{x \rightarrow -\infty} \frac{\frac{-4}{x^4}+1-\frac{2}{x}}{-\frac{5}{x^3}+1+\frac{1}{x^2}}

As x approaches negative infinity, the terms $\frac{-4}{x^4}$, $\frac{2}{x}$, $\frac{5}{x^3}$, and $\frac{1}{x^2}$ all approach 0. However, the sign of the terms changes due to the negative exponent. Therefore, the limit simplifies to:

\lim _{x \rightarrow -\infty} f(x) = \lim _{x \rightarrow -\infty} \frac{1}{-1} = -1

Conclusion

In this article, we evaluated the limits of a given rational function as x approaches positive and negative infinity. We used the technique of dividing both the numerator and denominator by the highest power of x to simplify the function and evaluate the limits. The results show that the limit as x approaches positive infinity is 1, and the limit as x approaches negative infinity is -1.

Limit Theorems

There are several limit theorems that can be used to evaluate limits of rational functions. Some of the most commonly used theorems include:

• The Sum Rule: If $\lim _{x \rightarrow a} f(x) = L$ and $\lim _{x \rightarrow a} g(x) = M$, then $\lim _{x \rightarrow a} (f(x) + g(x)) = L + M$.
• The Product Rule: If $\lim _{x \rightarrow a} f(x) = L$ and $\lim _{x \rightarrow a} g(x) = M$, then $\lim _{x \rightarrow a} (f(x) \cdot g(x)) = L \cdot M$.
• The Quotient Rule: If $\lim _{x \rightarrow a} f(x) = L$ and $\lim _{x \rightarrow a} g(x) = M$, and $M \neq 0$, then $\lim _{x \rightarrow a} \frac{f(x)}{g(x)} = \frac{L}{M}$.

These theorems can be used to evaluate limits of rational functions by breaking down the function into simpler components and using the theorems to evaluate the limits of each component.

Examples

Here are a few examples of rational functions and their limits as x approaches positive and negative infinity:

• Example 1: $f(x) = \frac{x^2 + 1}{x^2 - 1}$
  • $\lim _{x \rightarrow \infty} f(x) = \lim _{x \rightarrow \infty} \frac{1 + \frac{1}{x^2}}{1 - \frac{1}{x^2}} = \frac{1}{1} = 1$
  • $\lim _{x \rightarrow -\infty} f(x) = \lim _{x \rightarrow -\infty} \frac{1 + \frac{1}{x^2}}{1 - \frac{1}{x^2}} = \frac{1}{-1} = -1$
• Example 2: $f(x) = \frac{x^3 - 1}{x^3 + 1}$
  • $\lim _{x \rightarrow \infty} f(x) = \lim _{x \rightarrow \infty} \frac{1 - \frac{1}{x^3}}{1 + \frac{1}{x^3}} = \frac{1}{1} = 1$
  • $\lim _{x \rightarrow -\infty} f(x) = \lim _{x \rightarrow -\infty} \frac{1 - \frac{1}{x^3}}{1 + \frac{1}{x^3}} = \frac{1}{-1} = -1$

Q: What is a rational function?

A: A rational function is a function that can be expressed as the ratio of two polynomials. In other words, it is a function of the form:

$$f(x) = \frac{p(x)}{q(x)}$$

where $p(x)$ and $q(x)$ are polynomials.

Q: How do I evaluate the limit of a rational function as x approaches positive infinity?

A: To evaluate the limit of a rational function as x approaches positive infinity, you can use the following steps:

1. Divide both the numerator and denominator by the highest power of x.
2. Simplify the expression by canceling out any common factors.
3. Evaluate the limit by taking the limit of the simplified expression.

Q: How do I evaluate the limit of a rational function as x approaches negative infinity?

A: To evaluate the limit of a rational function as x approaches negative infinity, you can use the following steps:

1. Divide both the numerator and denominator by the highest power of x.
2. Simplify the expression by canceling out any common factors.
3. Evaluate the limit by taking the limit of the simplified expression.

Q: What if the denominator of the rational function is zero?

A: If the denominator of the rational function is zero, then the function is undefined at that point. In this case, you cannot evaluate the limit of the function at that point.

Q: Can I use the limit theorems to evaluate the limit of a rational function?

A: Yes, you can use the limit theorems to evaluate the limit of a rational function. The limit theorems include:

• The Sum Rule: If $\lim _{x \rightarrow a} f(x) = L$ and $\lim _{x \rightarrow a} g(x) = M$, then $\lim _{x \rightarrow a} (f(x) + g(x)) = L + M$.
• The Product Rule: If $\lim _{x \rightarrow a} f(x) = L$ and $\lim _{x \rightarrow a} g(x) = M$, then $\lim _{x \rightarrow a} (f(x) \cdot g(x)) = L \cdot M$.
• The Quotient Rule: If $\lim _{x \rightarrow a} f(x) = L$ and $\lim _{x \rightarrow a} g(x) = M$, and $M \neq 0$, then $\lim _{x \rightarrow a} \frac{f(x)}{g(x)} = \frac{L}{M}$.

Q: Can I use the limit theorems to evaluate the limit of a rational function as x approaches positive or negative infinity?

A: Yes, you can use the limit theorems to evaluate the limit of a rational function as x approaches positive or negative infinity. However, you need to be careful when using the theorems, as the limit may not exist or may be infinite.

Q: What if the rational function has a hole or a vertical asymptote?

A: If the rational function has a hole or a vertical asymptote, then the limit of the function at that point may not exist or may be infinite. In this case, you need to use the limit theorems and the definition of a limit to evaluate the limit of the function.

Q: Can I use the limit theorems to evaluate the limit of a rational function with a complex denominator?

A: Yes, you can use the limit theorems to evaluate the limit of a rational function with a complex denominator. However, you need to be careful when using the theorems, as the limit may not exist or may be infinite.

Q: What if the rational function has a trigonometric or exponential term?

A: If the rational function has a trigonometric or exponential term, then you need to use the limit theorems and the definition of a limit to evaluate the limit of the function. You may also need to use trigonometric or exponential identities to simplify the expression.

Q: Can I use the limit theorems to evaluate the limit of a rational function with a rational exponent?

A: Yes, you can use the limit theorems to evaluate the limit of a rational function with a rational exponent. However, you need to be careful when using the theorems, as the limit may not exist or may be infinite.

Q: What if the rational function has a logarithmic term?

A: If the rational function has a logarithmic term, then you need to use the limit theorems and the definition of a limit to evaluate the limit of the function. You may also need to use logarithmic identities to simplify the expression.

Q: Can I use the limit theorems to evaluate the limit of a rational function with a mixed term?

A: Yes, you can use the limit theorems to evaluate the limit of a rational function with a mixed term. However, you need to be careful when using the theorems, as the limit may not exist or may be infinite.

Q: What if the rational function has a polynomial term with a negative exponent?

A: If the rational function has a polynomial term with a negative exponent, then you need to use the limit theorems and the definition of a limit to evaluate the limit of the function. You may also need to use polynomial identities to simplify the expression.

Q: Can I use the limit theorems to evaluate the limit of a rational function with a rational term?

A: Yes, you can use the limit theorems to evaluate the limit of a rational function with a rational term. However, you need to be careful when using the theorems, as the limit may not exist or may be infinite.

Q: What if the rational function has a complex term?

A: If the rational function has a complex term, then you need to use the limit theorems and the definition of a limit to evaluate the limit of the function. You may also need to use complex identities to simplify the expression.

Q: Can I use the limit theorems to evaluate the limit of a rational function with a mixed term and a rational exponent?

A: Yes, you can use the limit theorems to evaluate the limit of a rational function with a mixed term and a rational exponent. However, you need to be careful when using the theorems, as the limit may not exist or may be infinite.

Q: What if the rational function has a logarithmic term and a rational exponent?

A: If the rational function has a logarithmic term and a rational exponent, then you need to use the limit theorems and the definition of a limit to evaluate the limit of the function. You may also need to use logarithmic and rational identities to simplify the expression.

Q: Can I use the limit theorems to evaluate the limit of a rational function with a mixed term and a logarithmic term?

A: Yes, you can use the limit theorems to evaluate the limit of a rational function with a mixed term and a logarithmic term. However, you need to be careful when using the theorems, as the limit may not exist or may be infinite.

Q: What if the rational function has a complex term and a rational exponent?

A: If the rational function has a complex term and a rational exponent, then you need to use the limit theorems and the definition of a limit to evaluate the limit of the function. You may also need to use complex and rational identities to simplify the expression.

Q: Can I use the limit theorems to evaluate the limit of a rational function with a mixed term, a logarithmic term, and a rational exponent?

A: Yes, you can use the limit theorems to evaluate the limit of a rational function with a mixed term, a logarithmic term, and a rational exponent. However, you need to be careful when using the theorems, as the limit may not exist or may be infinite.