# How do you find the limit of # (x−pi/4)tan(2x) # as x approaches pi/4?

Please see the explanation section, below.

I'd like to use the fundamental trigonometric limts, so I'll change the variable.

Nww we have been asked to find

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We have that

#lim_(xrarrpi/4) (x−pi/4)tan(2x)= lim_(xrarrpi/4) (x-pi/4)*sin(2x)/(cos2x)#

#lim_(xrarrpi/4) (d[(x-pi/4)*sin2x]/dx)/(d(cos2x)/dx)=
lim_(xrarrpi/4) [sin2x+2*(x-pi/4)cos2x]/(-2sin2x)=-1/2#

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To find the limit of (x - pi/4)tan(2x) as x approaches pi/4, we can use the limit properties and trigonometric identities. By substituting pi/4 into the expression, we get (pi/4 - pi/4)tan(2(pi/4)), which simplifies to 0*tan(pi/2). Since the tangent function is undefined at pi/2, we need to further simplify the expression. By using the limit property that states if f(x) approaches 0 and g(x) is bounded, then the limit of f(x)*g(x) is 0, we can conclude that the limit of (x - pi/4)tan(2x) as x approaches pi/4 is 0.

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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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