Answer:
A and D
Step-by-step explanation:
Limit at positive infinity
[tex] \displaystyle \large{ \lim_{x \to \infty} {a}^{x} = \infty \: \: \: (a > 1)} \\ \displaystyle \large{ \lim_{x \to \infty} {a}^{x} = 1 \: \: \: (a = 1)} \\ \displaystyle \large{ \lim_{x \to \infty} {a}^{x} = 0 \: \: \: ( |a| < 1)} \\ \displaystyle \large{ \lim_{x \to \infty} {a}^{x} = no \: \: \: limit \: \: \: (a \leqslant - 1)} \\[/tex]
Limit at negative infinity
[tex] \displaystyle \large{ \lim_{x \to - \infty} {a}^{x} = 0 \: \: \: (a > 1)} \\ \displaystyle \large{ \lim_{x \to - \infty} {a}^{x} = 1 \: \: \: (a = 1)} \\ \displaystyle \large{ \lim_{x \to - \infty} {a}^{x} = \infty \: \: \: ( |a| < 1)} \\ \displaystyle \large{ \lim_{x \to - \infty} {a}^{x} = no \: \: \: limit \: \: \: (a \leqslant - 1)} \\[/tex]
Derived from Infinite Geometric Sequence.
For a>1, when x approaches infinite, y approaches infinite too.
For a=1, when x approaches infinite, y approaches 1.
For |a| < 1 or 0 < a < 1, when x approaches infinite, y approaches 0.
For a ≤ -1, when x approaches infinite, y approaches both infinity and negative infinity but since lim + ≠ lim - then it does not exist.
However, the explanation above is limit which is calculus so it may be advanced.
From the graph, x keeps increasing then y keeps increasing too and when x keeps decreasing, y keeps decreasing almost to 0.
