Understanding Growth at Infinity

To Find the limit as x→∞ of a rational function, we compare the degrees of the numerator and denominator. Here, both the numerator and denominator are polynomials of degree 2. When the degrees are equal, the limit is the ratio of the leading coefficients. The leading coefficient of the numerator is 3, and the leading coefficient of the denominator is 2. Therefore, the limit is 3/2.

To Find the limit as x→∞ of a rational function, we compare the degrees of the numerator and denominator. Here, the numerator is a polynomial of degree 1, and the denominator is a polynomial of degree 2. When the degree of the numerator is less than the degree of the denominator, the limit is 0. This is because the denominator grows much faster than the numerator as x approaches infinity, making the fraction approach 0.

To Find the limit as x→∞ of a rational function, we compare the degrees of the numerator and denominator. Here, both the numerator and denominator are polynomials of degree 3. When the degrees are equal, the limit is the ratio of the leading coefficients. The leading coefficient of the numerator is 1, and the leading coefficient of the denominator is 4. Therefore, the limit is 1/4.

To Find the limit as x→∞ of a rational function, we compare the degrees of the numerator and denominator. Here, the numerator is a polynomial of degree 2, and the denominator is a polynomial of degree 3. When the degree of the numerator is less than the degree of the denominator, the limit is 0. This is because the denominator grows much faster than the numerator as x approaches infinity, making the fraction approach 0.

To Find the limit as x→∞ of a rational function, we compare the degrees of the numerator and denominator. Here, both the numerator and denominator are polynomials of degree 1. When the degrees are equal, the limit is the ratio of the leading coefficients. The leading coefficient of the numerator is 3, and the leading coefficient of the denominator is 2. Therefore, the limit is 3/2.

To Find the limit as x→∞ of √(x² + 4)/x, we factor out x² from inside the square root: √(x²(1 + 4/x²))/x. Recall that √(x²) = |x|. Since x approaches positive infinity, |x| = x. The expression simplifies to x√(1 + 4/x²)/x, which reduces to √(1 + 4/x²). As x approaches infinity, 4/x² approaches 0, so the limit is 1.

To Find the limit as x→∞ of a rational function, we compare the degrees of the numerator and denominator. Here, the numerator is a polynomial of degree 2, and the denominator is a polynomial of degree 1. When the degree of the numerator is greater than the degree of the denominator, the limit is infinity (or negative infinity, depending on the signs of the leading coefficients). In this case, both leading coefficients are positive, so the limit is infinity.

To Find the limit as x→∞ of a rational function, we compare the degrees of the numerator and denominator. Here, both the numerator and denominator are polynomials of degree 3. When the degrees are equal, the limit is the ratio of the leading coefficients. The leading coefficient of the numerator is 2, and the leading coefficient of the denominator is 3. Therefore, the limit is 2/3.

To Find the limit as x→∞ of a rational function, we compare the degrees of the numerator and denominator. Here, the numerator is a polynomial of degree 1, and the denominator is a polynomial of degree 2. When the degree of the numerator is less than the degree of the denominator, the limit is 0. This is because the denominator grows much faster than the numerator as x approaches infinity, making the fraction approach 0.

To Find the limit as x→∞ of a rational function, we compare the degrees of the numerator and denominator. Here, both the numerator and denominator are polynomials of degree 2. When the degrees are equal, the limit is the ratio of the leading coefficients. The leading coefficient of the numerator is 4, and the leading coefficient of the denominator is 2. Therefore, the limit is 4/2 = 2.

When the limit as x approaches infinity of a function f(x) equals a finite number L, it means that as x gets larger and larger, the values of f(x) get closer and closer to L. Graphically, this means that the graph of f(x) approaches the horizontal line y = L as x moves to the right. This line y = L is called a horizontal asymptote of the function.

For rational functions (polynomials divided by polynomials), the limit as x approaches infinity depends on the degrees of the numerator and denominator. If the degree of the numerator is less than the degree of the denominator, the limit is 0. If the degree of the numerator equals the degree of the denominator, the limit is the ratio of the leading coefficients. If the degree of the numerator is greater than the degree of the denominator, the limit is infinity or negative infinity. Additionally, limits at infinity exist for many types of functions, not just polynomials.

When the limit as x approaches infinity of a function f(x) equals infinity, it means that as x gets larger and larger, the values of f(x) also get larger and larger without any upper bound. In other words, the function grows without bound as x increases. This is different from having a horizontal asymptote, which would occur if the limit was a finite number. It's also different from having a vertical asymptote, which relates to the behavior of the function as x approaches a finite value, not infinity.

If the limit as x approaches infinity of f(x)/g(x) = 0, it means that as x gets larger and larger, the value of f(x) becomes negligible compared to g(x). In other words, g(x) grows much faster than f(x). This is why their ratio approaches 0. If f(x) grew faster than g(x), the ratio would approach infinity. If they grew at the same rate, the ratio would approach a non-zero constant.

A function has a horizontal asymptote if the limit as x approaches infinity (or negative infinity) is a finite number. For option A, f(x) = x² + 3x + 2, the limit as x approaches infinity is infinity, so there is no horizontal asymptote. For option B, f(x) = 2^x, the limit as x approaches infinity is also infinity, so there is no horizontal asymptote. For option C, f(x) = (3x² + 1)/(x² + 4), the limit as x approaches infinity is 3 (the ratio of the leading coefficients), so there is a horizontal asymptote at y = 3. For option D, f(x) = ln(x), the limit as x approaches infinity is infinity, so there is no horizontal asymptote.