# How do you verify whether rolle's theorem can be applied to the function #f(x)=tanx# in [0,pi]?

When we are asked whether some theorem "can be applied" to some situation, we are really being asked "Are the hypotheses of the theorem true for this situation?"

(The hypotheses are also called the antecedent, of 'the if parts'.)

So we need to determine whether the hypotheses ot Rolle's Theorem are true for the function

Rolle's Theorem has three hypotheses:

We say that we can apply Rolle's Theorem if all 3 hypotheses are true.

At this point we could stop. Not all of the hypotheses are true. Let's look at the others for completeness.

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To verify whether Rolle's Theorem can be applied to the function ( f(x) = \tan(x) ) in the interval ([0, \pi]), we need to check the following conditions:

**Continuity**: ( f(x) ) must be continuous on the closed interval ([0, \pi]).**Differentiability**: ( f(x) ) must be differentiable on the open interval ((0, \pi)).**Boundary conditions**: ( f(0) = f(\pi) ).

For the function ( f(x) = \tan(x) ) in the interval ([0, \pi]):

**Continuity**: The function ( f(x) = \tan(x) ) is continuous on the interval ([0, \pi]) except at ( x = \frac{\pi}{2} ) due to a vertical asymptote.**Differentiability**: The function ( f(x) = \tan(x) ) is differentiable on the open interval ((0, \pi)) since it is continuous on that interval.**Boundary conditions**: ( f(0) = 0 ) and ( f(\pi) = 0 ), satisfying the boundary conditions.

Since all the conditions of Rolle's Theorem are satisfied for ( f(x) = \tan(x) ) in the interval ([0, \pi]), we can apply Rolle's Theorem to this function on this interval.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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