# How do you find the limit as x approaches #pi/4# of #[sin(x) - cos(x)] / cos(2x)#?

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If you want to evaluate the limit without l'Hopital's Rule (find the limit "algebraically"), do this:

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To find the limit as x approaches pi/4 of [sin(x) - cos(x)] / cos(2x), we can use algebraic manipulation and trigonometric identities.

First, we simplify the expression by expanding cos(2x) using the double angle formula: cos(2x) = cos^2(x) - sin^2(x).

Next, we substitute this expression back into the original equation: [sin(x) - cos(x)] / (cos^2(x) - sin^2(x)).

Now, we can factor out a common factor of (sin(x) - cos(x)) from the numerator: (sin(x) - cos(x)) / [(cos(x) + sin(x))(cos(x) - sin(x))].

Since sin(x) - cos(x) is a common factor in both the numerator and denominator, we can cancel it out: 1 / (cos(x) + sin(x)).

Finally, we substitute x = pi/4 into the expression: 1 / (cos(pi/4) + sin(pi/4)).

Evaluating cos(pi/4) and sin(pi/4), we get: 1 / (√2/2 + √2/2) = 1 / √2 = √2/2.

Therefore, the limit as x approaches pi/4 of [sin(x) - cos(x)] / cos(2x) is √2/2.

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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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