lim as x heads toward infinity of x^200/e^.02x
To find the limit of the function as x approaches infinity, we can use L'Hôpital's rule, which is a method of finding limits of indeterminate forms of the type "∞/∞" or "0/0".
First, let's rewrite the function in a suitable form:
f(x) = x^200 / e^(0.02x)
As x approaches infinity, the denominator, e^(0.02x), grows much faster than the numerator, x^200. This suggests that the function will approach zero as x heads towards infinity.
Now, we can apply L'Hôpital's rule to find the limit:
1. Take the derivative of the numerator and the denominator:
f'(x) = 200x^199
g'(x) = 0.02e^(0.02x)
2. Now, calculate the limit of the derivatives as x approaches infinity:
lim as x → ∞ f'(x) = lim as x → ∞ 200x^199 = ∞
lim as x → ∞ g'(x) = lim as x → ∞ 0.02e^(0.02x) = 0.02e^∞ = ∞
3. Apply L'Hôpital's rule by taking the ratio of the derivatives:
lim as x → ∞ f'(x) / g'(x) = (∞) / (∞) = 1
Since the limit of the derivative ratio is 1, the limit of the original function f(x) as x approaches infinity is also 1.
Therefore, lim as x → ∞ x^200 / e^(0.02x) = 1.