True or False
If lim x→∞ f(x) = 1and lim x→∞ g(x) = ∞,then lim x→∞ [f(x)]^g(x) = 1.
false
think of the definition of e: lim(x->∞) (1 + 1/x)^x
To determine whether the statement is true or false, we can use the property of limits.
We are given that the limit of f(x) as x approaches infinity is 1, and the limit of g(x) as x approaches infinity is infinity. We want to find the limit of [f(x)]^g(x) as x approaches infinity.
Let's analyze the possible scenarios:
1. If g(x) approaches infinity faster than f(x) approaches 1, then [f(x)]^g(x) will become infinite as x approaches infinity.
2. If f(x) approaches 1 faster than g(x) approaches infinity, then [f(x)]^g(x) will approach 0 as x approaches infinity.
3. If f(x) and g(x) approach their respective limits at a balanced rate, then we may have an indeterminate form.
Since none of the above scenarios result in a limit of 1, we can conclude that the statement is false.
To understand this conceptually, consider an example where f(x) approaches 1, but g(x) grows much faster. For instance, let's say f(x) = 1 + 1/x and g(x) = x. As x approaches infinity, f(x) approaches 1, but g(x) grows infinitely larger. In this case, [f(x)]^g(x) will approach 0, not 1.