Question

Use your calculator to select the best answer below:

limit as x goes to infinity of the quotient of 10 raised to the x power and the quantity negative e raised to the x power
a. 1
b. -1
c. 0
d. does not exist

Answer 1

The limit of 10 to the power of x divided by negative e to the power of x as x approaches infinity is negative 1. This is due to the faster growth rate of the numerator, 10^x, over the denominator, -e^x, considering both the values and the negative sign.

Step-by-step explanation:

The limit of 10x divided by -ex as x approaches infinity is what we are trying to determine. By using a property of limits for exponential functions, we can predict how the function behaves as x increases without bound.

Since the base e is approximately 2.718, and 10 is greater than e, 10x will grow at a faster rate than ex. However, because we are dividing by -ex, the negative sign will ensure that the limit is -1.

When dealing with limits of this nature, the growth rates of the numerator and the denominator are what determine the outcome.

The limit approaches -1 because the growth rate of the numerator outweighs the growth rate of the denominator by a constant multiplier and the negative sign from the denominator dictates that the result will be negative.