Question

find the limit. (if the limit is infinite, enter '[infinity]' or '-[infinity]', as appropriate. if the limit does not otherwise exist, enter dne.) lim x→[infinity] 5 − ex 5 8ex

Answer 1

The limit of the expression as x approaches infinity simplifies down to a value of -1/40, after the exponential terms cancel out.

Step-by-step explanation:

The question involves finding the limit of a function as x approaches infinity. Specifically, we are tasked to evaluate the limit of the expression 5 - ex / (5 * 8ex) as x approaches infinity. To find this limit, we can analyze the behavior of the numerator and the denominator separately as x grows larger and larger.

As x approaches infinity, ex, which is the exponential function, grows at a much faster rate compared to any polynomial or constant in the expression. Therefore, both the numerator and the denominator will be dominated by the exponential terms. In the numerator, ex will make the constant 5 insignificant, and in the denominator, 5 * 8ex will effectively become 40ex. Thus, the dominant terms in the limit are -ex in the numerator and 40ex in the denominator.

Simplifying, the expression becomes -1/40 as x approaches infinity, since ex cancels out. The limit is therefore -1/40, which is a finite number.