The limit as x approaches infinity. (1)/(5^x)

The limit as x approaches 1. (1-x^3)/(2-sqrt(x^2-3))

Show your work thanks in advance!

what do you think? Both are straighforward, with the caveat the second has a complex number limit(it has a sqrt (-2) in the denominator.

how is it straightforward? they both go to indeterminates.

Nope, the first has real number in the numerator, and a infinity in the denominator.

The second has a zero in the numerator, and a complex number in the denominator.

To find the limits of the given expressions, we can use basic limit properties and algebraic manipulations.

1. Limit as x approaches infinity of (1)/(5^x):
As x approaches infinity, the value of 5^x increases without bound, becoming larger and larger. Thus, the value of the expression (1)/(5^x) approaches 0. This can be shown more formally as follows:

lim(x→∞) (1)/(5^x)
= 1/lim(x→∞) (5^x)
= 1/∞
= 0

Therefore, the limit as x approaches infinity of (1)/(5^x) is 0.

2. Limit as x approaches 1 of (1-x^3)/(2-sqrt(x^2-3)):
To evaluate this limit, we substitute x = 1 into the expression and simplify:

lim(x→1) (1-x^3)/(2-sqrt(x^2-3))
= (1-1^3)/(2-sqrt(1^2-3))
= 0/(2-sqrt(1-3))
= 0/(2-sqrt(-2))

At this point, we encounter a problem with the square root of a negative number. Since we cannot take the square root of a negative number in the real number system, the expression is undefined. Therefore, the limit as x approaches 1 of (1-x^3)/(2-sqrt(x^2-3)) does not exist.

In summary:
- The limit as x approaches infinity of (1)/(5^x) is 0.
- The limit as x approaches 1 of (1-x^3)/(2-sqrt(x^2-3)) does not exist.