The end behavior of the logarithmic function f(x) = log3(x2) can be analyzed by considering its limits as x approaches both positive and negative infinity.
First, as x approaches infinity (x → ∞), we observe that:
- As x becomes larger, x2 also becomes larger. Therefore, the logarithm of x2 will also increase.
- This means that f(x) = log3(x2) will approach infinity.
In mathematical terms, we can say:
lim (x → ∞) f(x) = ∞
Next, let’s consider x approaching zero from the right (x → 0+):
- When x is very small but positive, x2 is also small and positive.
- As the input to the log function approaches 0, the logarithm function itself will decrease without bound.
Thus, we can express this as:
lim (x → 0+) f(x) = -∞
Finally, it’s important to note that logarithmic functions are not defined for non-positive values. Therefore, f(x) = log3(x2) does not exist for x ≤ 0.
In summary:
- As x → ∞, f(x) → ∞.
- As x → 0+, f(x) → -∞.
- The function is only defined for positive x.
This understanding of the end behavior is crucial for applications in calculus and real-world problems where logarithmic growth is examined.