What simpler function models the end behavior of the rational function in #4?
#4: lim x->7 (sqrt(x+2)-3)/(x-7)
as x→∞, so do x+2 and x-7. So they are equivalent to just x. So the limit is the same as 7√x
To determine the end behavior of the given rational function, we need to provide a simpler function that captures the same behavior as x approaches positive infinity and negative infinity.
First, let's simplify the given rational function:
f(x) = (sqrt(x + 2) - 3)/(x - 7)
As x approaches positive infinity, both the numerator and the denominator of the function tend to infinity. In this case, we can simplify the function by dividing both the numerator and denominator by x, considering the term with the highest degree in the denominator.
Simplifying the function, we get:
f(x) = (sqrt(x + 2)/x - 3/x) / (x/x - 7/x)
= sqrt(x)/x - 3/x / (1 - 7/x)
≈ sqrt(x)/x - 3/x / (1 - 0)
≈ sqrt(x)/x - 3/x
Now, as x approaches positive infinity, the terms with x in the denominator become insignificant compared to the term sqrt(x)/x. Therefore, we can simplify the function as x approaches positive infinity:
f(x) ≈ sqrt(x)/x
Similarly, as x approaches negative infinity, both the numerator and the denominator of the function tend to negative infinity. Again, we can simplify the function by dividing both the numerator and denominator by x, considering the term with the highest degree in the denominator.
Simplifying the function, we get:
f(x) = (sqrt(x + 2)/x - 3/x) / (x/x - 7/x)
= sqrt(x)/x - 3/x / (1 - 7/x)
≈ sqrt(x)/x - 3/x / (1 - 0)
≈ sqrt(x)/x - 3/x
Now, as x approaches negative infinity, the terms with x in the denominator become insignificant compared to the term sqrt(x)/x. Therefore, we can simplify the function as x approaches negative infinity:
f(x) ≈ sqrt(x)/x
Therefore, a simpler function that models the end behavior of the given rational function is f(x) ≈ sqrt(x)/x.