Evaluate the following limits.
lim as x approaches infinity 6/e^x + 7=____?
lim as x approaches negative infinity 6/e^x+7=____?
for the first
as x gets larger, 6/e^x gets closer to zero
so the limit is 0+7 = 7
for the second, remember that 1/a^-n = a^n
so for huge negative values of x, 6/e^x becomes "huge"
so for the second, it is undefined (infinitely large)
To evaluate the given limits, we can use the properties of exponential functions and the rules of limits.
First, let's evaluate the limit as x approaches infinity:
lim (x->∞) (6/e^x + 7)
As x approaches infinity, the value of e^x also approaches infinity. Since the denominator of the fraction is e^x, it becomes larger and larger, approaching infinity as well. Therefore, the fraction 6/e^x approaches 0.
lim (x->∞) (6/e^x + 7) = 0 + 7 = 7.
So, the first limit is equal to 7.
Next, let's evaluate the limit as x approaches negative infinity:
lim (x->-∞) (6/e^x + 7)
As x approaches negative infinity, the value of e^x approaches 0. Since the denominator of the fraction is e^x, it becomes smaller and smaller, approaching 0. Therefore, the fraction 6/e^x approaches positive infinity (∞).
lim (x->-∞) (6/e^x + 7) = ∞ + 7 = ∞.
So, the second limit is equal to positive infinity (∞).
In summary:
lim (x->∞) (6/e^x + 7) = 7
lim (x->-∞) (6/e^x + 7) = ∞