To determine the correct statement, let's analyze the behavior of the exponential function [tex]\( f(x) \)[/tex] and the logarithmic function [tex]\( g(x) \)[/tex] as [tex]\( x \rightarrow \infty \)[/tex].
1. Exponential Function [tex]\( f(x) \)[/tex]:
- An exponential function typically takes the form [tex]\( f(x) = a^x \)[/tex] where [tex]\( a > 1 \)[/tex].
- As [tex]\( x \)[/tex] approaches infinity ([tex]\( x \rightarrow \infty \)[/tex]), the value of [tex]\( f(x) \)[/tex] grows significantly larger because [tex]\( a^x \)[/tex] increases rapidly.
2. Logarithmic Function [tex]\( g(x) \)[/tex]:
- A logarithmic function typically takes the form [tex]\( g(x) = \log_b(x) \)[/tex] where [tex]\( b > 1 \)[/tex].
- As [tex]\( x \)[/tex] approaches infinity ([tex]\( x \rightarrow \infty \)[/tex]), the value of [tex]\( g(x) \)[/tex] also increases, albeit more slowly than the exponential function.
Let's evaluate the provided statements based on these observations:
1. As [tex]\( x \rightarrow \infty, f(x) \rightarrow \infty \)[/tex] and [tex]\( g(x) \rightarrow \infty \)[/tex]:
- This statement is correct because both the exponential function [tex]\( f(x) \)[/tex] and the logarithmic function [tex]\( g(x) \)[/tex] approach infinity as [tex]\( x \)[/tex] grows larger.
2. As [tex]\( x \rightarrow \infty, f(x) \rightarrow \infty \)[/tex] and [tex]\( g(x) \rightarrow 1 \)[/tex]:
- This statement is incorrect. While [tex]\( f(x) \)[/tex] does approach infinity, [tex]\( g(x) \)[/tex] does not approach 1 but instead grows larger and tends towards infinity.
3. As [tex]\( x \rightarrow \infty, f(x) \rightarrow 3 \)[/tex] and [tex]\( g(x) \rightarrow 1 \)[/tex]:
- This statement is incorrect. Neither [tex]\( f(x) \)[/tex] approaches a constant value, nor does [tex]\( g(x) \)[/tex], as both grow towards infinity.
4. As [tex]\( x \rightarrow \infty, f(x) \rightarrow 3 \)[/tex] and [tex]\( g(x) \rightarrow \infty \)[/tex]:
- This statement is incorrect. While [tex]\( g(x) \)[/tex] does approach infinity, [tex]\( f(x) \)[/tex] does not approach a constant value but instead grows larger.
After analyzing these behaviors, the correct statement is:
As [tex]\( x \rightarrow \infty, f(x) \rightarrow \infty \)[/tex] and [tex]\( g(x) \rightarrow \infty \)[/tex].
So the correct answer is:
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