For which pair of functions f(x) and g(x) below will the lim f(x)/g(x)equal 0
x-infinity
f(x) e^x; g(x) = x^3
f(x) x^5; g(x) = e^x
f(x) x^3; g(x) = ln(x)
f(x) x^negative 2; g(x) = e^negative x
To determine which pair of functions will make the limit of f(x)/g(x) equal to 0 as x approaches negative infinity, we must evaluate the limits of each pair.
1. f(x) = e^x and g(x) = x^3
Taking the limit of f(x)/g(x) as x approaches negative infinity:
lim(x→-∞) e^x/x^3
The exponential function e^x grows faster than any power of x as x approaches infinity. Thus, the denominator x^3 grows significantly faster than the numerator e^x. As a result, the limit of this expression is 0.
2. f(x) = x^5 and g(x) = e^x
Taking the limit of f(x)/g(x) as x approaches negative infinity:
lim(x→-∞) x^5/e^x
In this case, the exponential function e^x grows much faster than any power of x as x approaches negative infinity. Consequently, the denominator e^x grows much faster compared to the numerator x^5. Hence, the limit of this expression is 0.
3. f(x) = x^3 and g(x) = ln(x)
Taking the limit of f(x)/g(x) as x approaches negative infinity:
lim(x→-∞) x^3/ln(x)
The natural logarithm function ln(x) grows much slower than any power of x as x approaches infinity. Therefore, the denominator ln(x) grows significantly slower compared to the numerator x^3. Consequently, the limit of this expression is positive infinity, not 0.
4. f(x) = x^-2 and g(x) = e^(-x)
Taking the limit of f(x)/g(x) as x approaches negative infinity:
lim(x→-∞) x^(-2)/e^(-x)
Similar to previous cases, the exponential function e^(-x) grows much faster than any power of x as x approaches negative infinity. Hence, the denominator e^(-x) grows significantly faster than the numerator x^(-2). Thus, the limit of this expression is 0.
In summary, the pair of functions where the limit of f(x)/g(x) equals 0 as x approaches negative infinity is f(x) = x^(-2) and g(x) = e^(-x).