End Behavior of Functions Quiz

1) What is the limit as x→∞ of (x² + 3x + 2)/(2x² + 5x + 1)?

0

1/2

1

∞

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 1, and the leading coefficient of the denominator is 2. Therefore, the limit is 1/2.

Explanation

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 1, and the leading coefficient of the denominator is 2. Therefore, the limit is 1/2.

2) What is the limit as x→∞ of (3x³ + 2x)/(x³ + 4)?

0

3

1

∞

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 3, and the leading coefficient of the denominator is 1. Therefore, the limit is 3/1 = 3.

Explanation

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 3, and the leading coefficient of the denominator is 1. Therefore, the limit is 3/1 = 3.

3) What is the limit as x→∞ of (2x + 1)/(x² + 3x + 2)?

0

2

1/2

∞

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.

Explanation

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.

4) What is the limit as x→∞ of (x^4 + 2x²)/(3x^4 + 5x³ + 1)?

0

1/3

1

∞

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 4. 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 3. Therefore, the limit is 1/3.

Explanation

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 4. 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 3. Therefore, the limit is 1/3.

5) What is the limit as x→∞ of (x³ + 2)/(2x² + 3x)?

0

1/2

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 3, and the denominator is a polynomial of degree 2. 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.

Explanation

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 3, and the denominator is a polynomial of degree 2. 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.

6) What is the limit as x→∞ of √(9x² + 4)/x?

0

1

3

∞

To Find the limit as x→∞ of √(9x² + 4)/x, we can simplify the expression. First, we factor out x² from inside the square root: √(x²(9 + 4/x²))/x. This simplifies to x*√(9 + 4/x²)/x, which further simplifies to √(9 + 4/x²). As x approaches infinity, 4/x² approaches 0, so the expression approaches √(9 + 0) = 3.

Explanation

To Find the limit as x→∞ of √(9x² + 4)/x, we can simplify the expression. First, we factor out x² from inside the square root: √(x²(9 + 4/x²))/x. This simplifies to x*√(9 + 4/x²)/x, which further simplifies to √(9 + 4/x²). As x approaches infinity, 4/x² approaches 0, so the expression approaches √(9 + 0) = 3.

7) What is the limit as x→∞ of (x² + 5)/(2x + 3)?

0

1/2

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.

Explanation

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.

8) What is the limit as x→∞ of (4x⁵ + 2x³)/(2x⁵ + x^4 + 1)?

0

2

1/2

∞

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 5. 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.

Explanation

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 5. 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.

9) What is the limit as x→∞ of (3x + 2)/(x² + 4x + 1)?

0

3

1/3

∞

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.

Explanation

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.

10) What is the limit as x→∞ (5x³ + 2x²)/(x³ + 3x + 1)?

0

5

1/5

∞

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 5, and the leading coefficient of the denominator is 1. Therefore, the limit is 5/1 = 5.

Explanation

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 5, and the leading coefficient of the denominator is 1. Therefore, the limit is 5/1 = 5.

11) What does the limit as x approaches infinity of f(x) = 0 tell us about the function f(x)?

The function has a horizontal asymptote at y = 0.

The function has a vertical asymptote at x = 0.

The function approaches 0 as x approaches 0.

The function is constant at 0.

When the limit as x approaches infinity of a function f(x) equals 0, it means that as x gets larger and larger, the values of f(x) get closer and closer to 0. Graphically, this means that the graph of f(x) approaches the horizontal line y = 0 (the x-axis) as x moves to the right. This line y = 0 is called a horizontal asymptote of the function.

Explanation

When the limit as x approaches infinity of a function f(x) equals 0, it means that as x gets larger and larger, the values of f(x) get closer and closer to 0. Graphically, this means that the graph of f(x) approaches the horizontal line y = 0 (the x-axis) as x moves to the right. This line y = 0 is called a horizontal asymptote of the function.

12) Which of the following statements about limits at infinity is true?

If the degree of the numerator is less than the degree of the denominator, the limit is infinity.

If the degree of the numerator equals the degree of the denominator, the limit is 0.

Limits at infinity describe the end behavior of a function.

Limits at infinity indeed describe the end behavior of a function - what happens to the function values as x becomes very large (positively or negatively). The other options are incorrect: if the degree of the numerator is less than the degree of the denominator, the limit is 0, not infinity. If the degrees are equal, the limit is the ratio of the leading coefficients, not 0. If the degree of the numerator is greater, the limit is infinity or negative infinity, not the ratio of the leading coefficients.

Explanation

Limits at infinity indeed describe the end behavior of a function - what happens to the function values as x becomes very large (positively or negatively). The other options are incorrect: if the degree of the numerator is less than the degree of the denominator, the limit is 0, not infinity. If the degrees are equal, the limit is the ratio of the leading coefficients, not 0. If the degree of the numerator is greater, the limit is infinity or negative infinity, not the ratio of the leading coefficients.

13) What does the limit as x approaches infinity of f(x) = 3 tell us about the function f(x)?

The function has a vertical asymptote at x = 3.

The function has a horizontal asymptote at y = 3.

The function approaches 3 as x approaches 0.

The function is constant at 3.

When the limit as x approaches infinity of a function f(x) equals a finite number (in this case, 3), it means that as x gets larger and larger, the values of f(x) get closer and closer to 3. Graphically, this means that the graph of f(x) approaches the horizontal line y = 3 as x moves to the right. This line y = 3 is called a horizontal asymptote of the function.

Explanation

When the limit as x approaches infinity of a function f(x) equals a finite number (in this case, 3), it means that as x gets larger and larger, the values of f(x) get closer and closer to 3. Graphically, this means that the graph of f(x) approaches the horizontal line y = 3 as x moves to the right. This line y = 3 is called a horizontal asymptote of the function.

14) If the limit as x approaches infinity of f(x)/g(x) = 1, what can we conclude about the relative growth rates of f(x) and g(x)?

F(x) grows faster than g(x).

G(x) grows faster than f(x).

F(x) and g(x) grow at the same rate.

No conclusion can be drawn about their relative growth rates.

If the limit as x approaches infinity of f(x)/g(x) = 1, it means that as x gets larger and larger, the value of f(x) becomes approximately equal to g(x). In other words, f(x) and g(x) grow at the same rate. This is why their ratio approaches 1. If f(x) grew faster than g(x), the ratio would approach infinity. If g(x) grew faster than f(x), the ratio would approach 0.

Explanation

If the limit as x approaches infinity of f(x)/g(x) = 1, it means that as x gets larger and larger, the value of f(x) becomes approximately equal to g(x). In other words, f(x) and g(x) grow at the same rate. This is why their ratio approaches 1. If f(x) grew faster than g(x), the ratio would approach infinity. If g(x) grew faster than f(x), the ratio would approach 0.

15) Which of the following functions has a horizontal asymptote at y = 0?

F(x) = x + 1

F(x) = 1/x

F(x) = (x² + 1)/(x + 1)

F(x) = x²

A function has a horizontal asymptote at y = L if the limit as x approaches infinity of the function equals L. For option A, f(x) = x + 1, the limit as x approaches infinity is infinity, so there is no horizontal asymptote. For option B, f(x) = 1/x, the limit as x approaches infinity is 0, so there is a horizontal asymptote at y = 0. For option C, f(x) = (x² + 1)/(x + 1), the limit as x approaches infinity is infinity (since the degree of the numerator is greater than the degree of the denominator), so there is no horizontal asymptote. For option D, f(x) = x², the limit as x approaches infinity is infinity, so there is no horizontal asymptote.

Explanation

A function has a horizontal asymptote at y = L if the limit as x approaches infinity of the function equals L. For option A, f(x) = x + 1, the limit as x approaches infinity is infinity, so there is no horizontal asymptote. For option B, f(x) = 1/x, the limit as x approaches infinity is 0, so there is a horizontal asymptote at y = 0. For option C, f(x) = (x² + 1)/(x + 1), the limit as x approaches infinity is infinity (since the degree of the numerator is greater than the degree of the denominator), so there is no horizontal asymptote. For option D, f(x) = x², the limit as x approaches infinity is infinity, so there is no horizontal asymptote.