Final answer:
The end behavior of the function h(x) as x approaches negative infinity is a constant value. By analyzing the growth rates of the function's terms, we find that the limit of h(x) approaches -2/7.
Step-by-step explanation:
To determine the end behavior of the function h(x)=-2x-8/√(49x²-9) as x approaches negative infinity, we need to analyze the terms involved as x becomes very large negatively. First, consider the numerator -2x-8. As x approaches negative infinity, this term will approach positive infinity (since multiplying a negative by a negative gives a positive). Then, consider the radical in the denominator, √(49x²-9); the term 49x² is the dominant term since it grows much faster than 9 as x grows large in absolute value. Since x is negative, 49x² will be positive, and thus the entire radical will approach positive infinity as well.
Combining both observations, we have a situation where both the numerator and the denominator of h(x) grow towards infinity. We look at their growth rates in more detail: -2x-8 grows linearly, while √(49x²-9) grows at the rate of x (since the square root of x² is x), which means both are proportional to x. Therefore, as x approaches negative infinity, the ratio of numerator and denominator will approach a finite limit, since they grow at the same rate.
The limit of h(x) as x approaches negative infinity is then determined by the coefficients of x in the numerator and the denominator. Simplifying, we get:
lim (x→-infinity) -2x-8/√(49x²-9) = lim (x→-infinity) -2x/√(49x²) = lim (x→-infinity) -2x/(7x) = -2/7
Thus, the end behavior of h(x) as x approaches negative infinity is that it will approach the constant value of -2/7.