The tangent function, denoted by tan(x), does not have a limit as x approaches infinity. This is because the function oscillates infinitely between positive and negative infinity as x increases.
Here's why:
- Periodic Nature: The tangent function has a period of π, meaning its graph repeats itself every π units.
- Vertical Asymptotes: As x approaches odd multiples of π/2 (e.g., π/2, 3π/2, 5π/2), the tangent function approaches positive or negative infinity, creating vertical asymptotes.
- No Convergence: Due to these vertical asymptotes, the function doesn't settle down to a specific value as x approaches infinity. It continues to oscillate without a defined limit.
In simpler terms: Imagine a rollercoaster that keeps going up and down forever, never reaching a specific height. That's what the tangent function does as x gets larger and larger.
