Let's evaluate the limit as x approaches infinity of the fraction (20x squared minus 13x plus 5) divided by (5 minus 4x cubed). To solve this limit, we need to compare the degrees of the numerator and denominator. The numerator has degree 2, while the denominator has degree 3. When the degree of the denominator is greater than the degree of the numerator, and x approaches infinity, the limit equals zero. Therefore, the answer is option B: zero.
Another way to solve this limit is to divide both the numerator and denominator by the highest power of x in the denominator, which is x cubed. After dividing, we get the expression shown. Now we can evaluate the limit of each term as x approaches infinity. The limit of 20 over x is 0. The limit of 13 over x squared is 0. The limit of 5 over x cubed is 0. And the limit of negative 4 is negative 4. Substituting these values, we get 0 divided by negative 4, which equals 0. This confirms our answer is option B: zero.
Let's understand this limit graphically. For rational functions as x approaches infinity, the behavior depends on the relationship between the degrees of the numerator and denominator. When the degree of the numerator is less than the degree of the denominator, as in our case, the limit equals zero. This means the function approaches the horizontal line y equals zero as x gets very large. If the degrees were equal, the limit would be the ratio of the leading coefficients. If the degree of the numerator were greater, the limit would be infinity or negative infinity. In our problem, since the numerator has degree 2 and the denominator has degree 3, the function approaches zero as x approaches infinity.
Let's compare our limit with similar rational functions to better understand the pattern. In our original problem, the degree of the numerator is less than the degree of the denominator, so the limit equals zero. If we modify the function to have equal degrees in the numerator and denominator, such as x cubed in the numerator, the limit would equal the ratio of the leading coefficients, which is negative 5. If the degree of the numerator is greater than the denominator, such as x to the fourth power in the numerator, the limit would be infinity or negative infinity, depending on the signs of the leading terms. The graph shows how these three different types of rational functions behave as x approaches infinity. Our original function, shown in blue, approaches the horizontal line at y equals zero.
To summarize what we've learned: The limit of a rational function as x approaches infinity depends on the relationship between the degrees of the numerator and denominator. When the degree of the numerator is less than the degree of the denominator, as in our problem, the limit equals zero. We can solve such limits either by comparing the degrees directly or by dividing both the numerator and denominator by the highest power of x in the denominator. For our specific problem, since the degree of the numerator is 2 and the degree of the denominator is 3, the limit equals zero, making option B the correct answer. Understanding this pattern is essential for evaluating limits of rational functions as x approaches infinity.