Yes, the tangent function (tan) can be infinite. This occurs at specific points where the cosine function, which is the denominator in the definition of tangent (tan x = sin x / cos x), equals zero.
Understanding the Tangent Function
The tangent function is defined as the ratio of the sine and cosine functions: tan(x) = sin(x) / cos(x). Since the sine function is bounded between -1 and 1, the tangent function becomes infinite whenever the cosine function approaches zero.
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Points of Infinity: This happens at odd multiples of π/2 (π/2, 3π/2, 5π/2, etc.). At these points, the cosine is zero, leading to division by zero, resulting in an infinite value. The function approaches positive infinity from one side and negative infinity from the other side at these points. This is represented by vertical asymptotes on the graph of the tangent function.
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Range of the Tangent Function: The provided text explicitly states: "Therefore, the sine function, which can also take any value between -1 and 1, is sometimes divided by very small numbers, which can result in very large positive or negative values. This is why the range of the tangent function is all real numbers, or in other words, infinite."
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Limit as x approaches infinity: It's crucial to understand that limx→∞ tan(x) does not exist. The tangent function oscillates between positive and negative infinity as x approaches infinity, never settling on a single value. This is different from saying that tan(x) can equal infinity at certain points.
Examples
- tan(π/2) is undefined (approaches ±∞)
- tan(3π/2) is undefined (approaches ±∞)
- tan(x) approaches infinity as x approaches π/2 from the right.
- tan(x) approaches negative infinity as x approaches π/2 from the left.
The statement on Quora that "tan(infinity) = 1" is incorrect. There is no single value for tan(x) as x approaches infinity.
