How do you find the limit of #(1-tanx)/(sinx-cosx)# as #x->pi/4#?
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To find the limit of (1 - tanx)/(sinx - cosx) as x approaches pi/4, we can use algebraic manipulation and trigonometric identities.
First, let's simplify the expression by multiplying the numerator and denominator by the conjugate of the denominator, which is (sinx + cosx):
[(1 - tanx)/(sinx - cosx)] * [(sinx + cosx)/(sinx + cosx)]
Expanding this expression, we get:
[(sinx + cosx - tanxsinx - tanxcosx)/(sinx - cosx)]
Next, we can simplify further by factoring out sinx from the numerator:
[sinx * (1 - tanx) + cosx * (1 - tanx)] / (sinx - cosx)
Now, we can cancel out the common factor of (1 - tanx):
[sinx + cosx] / (sinx - cosx)
Using the trigonometric identity sinx = cos(pi/2 - x), we can rewrite the expression as:
[cos(pi/2 - x) + cosx] / [cos(pi/2 - x) - cosx]
Applying the trigonometric identity cos(A) - cos(B) = -2sin((A + B)/2)sin((A - B)/2), we can simplify further:
[-2sin((pi/4 + x)/2)sin((pi/4 - x)/2)] / [-2sin((pi/4 + x)/2)sin((pi/4 - x)/2)]
The sin((pi/4 + x)/2) and sin((pi/4 - x)/2) terms cancel out, leaving us with:
-1
Therefore, the limit of (1 - tanx)/(sinx - cosx) as x approaches pi/4 is -1.
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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.
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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