# How do you find the limit of #(1-tan(x)) / (sin (x) - cos (x)) # as x approaches 0?

You can rewrite this as

#(1-tan(x)) / (sin (x) - cos (x)) =[1-(sinx/cosx)]/[sinx-cosx]= [cosx-sinx]/[cosx*(sinx-cosx)]= -1/cosx#

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To find the limit of (1 - tan(x))/(sin(x) - cos(x)) as x approaches 0, we can use algebraic manipulation and trigonometric identities.

First, let's simplify the expression by multiplying both the numerator and denominator by the conjugate of the denominator, which is (sin(x) + cos(x)). This will help us eliminate the trigonometric functions in the denominator.

(1 - tan(x))/(sin(x) - cos(x)) * (sin(x) + cos(x))/(sin(x) + cos(x))

Expanding this expression, we get:

[(1 - tan(x))(sin(x) + cos(x))]/[(sin(x) - cos(x))(sin(x) + cos(x))]

Next, we can simplify the numerator using the distributive property:

sin(x) + cos(x) - tan(x)sin(x) - tan(x)cos(x)

Now, we can apply trigonometric identities to simplify further. Using the identity tan(x) = sin(x)/cos(x), we can rewrite the expression as:

sin(x) + cos(x) - (sin(x)/cos(x))sin(x) - (sin(x)/cos(x))cos(x)

Expanding and rearranging terms, we have:

sin(x) + cos(x) - sin^2(x)/cos(x) - sin(x)cos(x)/cos(x)

Combining like terms, we get:

sin(x) + cos(x) - sin^2(x)/cos(x) - sin(x)

Now, we can simplify the expression by factoring out sin(x) from the numerator:

sin(x)(1 - sin(x))/cos(x) - sin(x)

Finally, we can cancel out the common factor of sin(x) in the numerator and denominator:

(1 - sin(x))/cos(x) - 1

As x approaches 0, sin(x) approaches 0, so we can substitute sin(x) with 0 in the expression:

(1 - 0)/cos(0) - 1

Simplifying further, we have:

1/cos(0) - 1

Since cos(0) equals 1, the expression becomes:

1/1 - 1

Which simplifies to:

1 - 1

Therefore, the limit of (1 - tan(x))/(sin(x) - cos(x)) as x approaches 0 is 0.

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