lim x ->infinity (e^x/lnx)
I got infinity/0 but is that allowed to be my answer?
what about this one:
limx->infinity (sqrt of x /e^x)
i got 0/infinity for this.
on the first, L'Hopitals rule
lim= e^x/ (1/x)= xe^x inf
second: lim= 1/2 (1/sqrtx)*(1/e^x) looks like zero.
For the first one, is it infinity?
To evaluate the limit of a function as x approaches infinity, we need to consider the relative growth rates of the numerator and denominator. Let's address each of the limits separately:
1. lim x -> infinity (e^x/ln(x)):
To determine the limit as x approaches infinity, we examine the growth rates of the numerator and denominator. The exponential function e^x grows faster than the logarithmic function ln(x) as x tends to infinity. Hence, when we have an expression of the form e^x/ln(x), the numerator grows significantly faster than the denominator, leading to the limit being infinity.
Therefore, in the case of lim x -> infinity (e^x/ln(x)), your answer of infinity is correct.
2. lim x -> infinity (sqrt(x) / e^x):
Similar to the previous case, we analyze the growth rates of the numerator (sqrt(x)) and the denominator (e^x) as x approaches infinity. Here, the exponential function e^x grows much faster than the square root function sqrt(x).
Since the numerator grows at a slower rate than the denominator, we conclude that the limit is zero.
Thus, in the case of lim x -> infinity (sqrt(x) / e^x), your answer of 0/infinity simplifies to 0.
In both cases, your answers are correct!