Find the horizontal asymptote of the graph of the function. (If an answer does not exist, enter DNE.)
f(x) = 5x^2 + 1/
7x^3 − 2x^2 + 5
To find the horizontal asymptote of the graph of the function f(x) = (5x^2 + 1) / (7x^3 - 2x^2 + 5), we evaluate the limit as x approaches positive or negative infinity.
First, let's consider x approaching positive infinity.
When x approaches positive infinity, the highest power terms dominate the expression. So, we ignore all the lower power terms.
The highest power term in the numerator and denominator is x^2 and x^3, respectively. Dividing each term in the numerator and denominator by x^3, we get:
f(x) = (5x^2 / x^3 + 1 / x^3) / (7x^3 / x^3 - 2x^2 / x^3 + 5 / x^3)
Simplifying further, we have:
f(x) = (5 / x + 1 / x^3) / (7 - 2 / x + 5 / x^3)
As x approaches positive infinity, both 1 / x and 1 / x^3 approach zero. Therefore, we can ignore these terms in the limit.
Taking the limit of f(x) as x approaches positive infinity:
lim(x->∞) (5 / x + 1 / x^3) / (7 - 2 / x + 5 / x^3)
Since 5 / x and 2 / x approach zero, we have:
lim(x->∞) (0 + 0) / (7 - 0 + 0) = 0 / 7 = 0
Therefore, as x approaches positive infinity, the graph of the function f(x) approaches a horizontal asymptote at y = 0.
Next, let's consider x approaching negative infinity.
Using similar reasoning as above, we divide each term in the numerator and denominator by x^3:
f(x) = (5 / x + 1 / x^3) / (7 - 2 / x + 5 / x^3)
As x approaches negative infinity, both 1 / x and 1 / x^3 approach zero (in the negative direction). Therefore, these terms can be ignored in the limit.
Taking the limit of f(x) as x approaches negative infinity:
lim(x->-∞) (5 / x + 1 / x^3) / (7 - 2 / x + 5 / x^3)
Again, since 5 / x and 2 / x approach zero, we have:
lim(x->-∞) (0 + 0) / (7 - 0 + 0) = 0 / 7 = 0
Therefore, as x approaches negative infinity, the graph of the function f(x) also approaches a horizontal asymptote at y = 0.
In summary, the horizontal asymptote of the graph of the function f(x) is y = 0.
To find the horizontal asymptote of the graph of the function f(x) = (5x^2 + 1)/(7x^3 − 2x^2 + 5), we need to examine the behavior of the function as x approaches positive or negative infinity.
First, let's compare the degrees of the numerator and denominator. The degree of the numerator is 2 (since it is a quadratic function) and the degree of the denominator is 3 (since it is a cubic function).
When the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is located at y = 0 (the x-axis).
Therefore, the horizontal asymptote of the given function is y = 0.
here the fraction approaches 5x^2/7x^3 = 5/7x -> 0
If the denominator has higher degree than the numerator, the horizontal asymptote is always y=0.