How do you compute the limit of #(sinx-cosx)/cos(2x)# as #x->pi/4#?

Now to evaluate the limit

To compute the limit of (sinx - cosx) / cos(2x) as x approaches pi/4, we can use algebraic manipulation and trigonometric identities.

First, substitute x = pi/4 into the expression: (sin(pi/4) - cos(pi/4)) / cos(2(pi/4))

Next, simplify the trigonometric values: (√2/2 - √2/2) / cos(pi/2)

Since (√2/2 - √2/2) equals zero, the numerator becomes zero: 0 / cos(pi/2)

The cosine of pi/2 is zero, so the denominator is also zero: 0 / 0

At this point, we have an indeterminate form of 0/0. To evaluate this limit further, we can apply L'Hôpital's rule.

Differentiating the numerator and denominator separately, we get: lim (x->pi/4) [d/dx (sinx - cosx) / d/dx cos(2x)]

Differentiating sinx and cosx gives us: lim (x->pi/4) [cosx + sinx / -2sin(2x)]

Substituting x = pi/4 into the expression: (cos(pi/4) + sin(pi/4)) / -2sin(2(pi/4))

Simplifying the trigonometric values: (√2/2 + √2/2) / -2sin(pi/2)

The numerator becomes (√2/2 + √2/2) = √2. The sine of pi/2 is 1, so the denominator is -2.

Therefore, the limit is: √2 / -2

In conclusion, the limit of (sinx - cosx) / cos(2x) as x approaches pi/4 is -√2/2.

To compute the limit of (sinx - cosx) / cos(2x) as x approaches π/4, we can use the following steps:

1. Substitute π/4 for x in the expression: (sin(π/4) - cos(π/4)) / cos(2 * π/4).
2. Simplify the expression: Evaluate sin(π/4), cos(π/4), and cos(2 * π/4) to get numerical values.
3. Calculate the limit of the simplified expression.

sin(π/4) = √2/2 cos(π/4) = √2/2 cos(2 * π/4) = cos(π/2) = 0

Now, substitute these values into the expression:

(sin(π/4) - cos(π/4)) / cos(2 * π/4) = (√2/2 - √2/2) / 0

Since the denominator is zero, we need to further evaluate the limit. We can try approaching the limit from the left and the right sides of π/4.

Approaching from the left side: As x approaches π/4 from the left side, sin(x) and cos(x) both approach √2/2. So, the expression becomes 0/0.

Approaching from the right side: As x approaches π/4 from the right side, sin(x) and cos(x) both approach √2/2. So, the expression becomes 0/0.

This indicates that we have an indeterminate form. To evaluate the limit, we can use L'Hôpital's Rule, which states that if we have an indeterminate form of 0/0 or ∞/∞, we can take the derivative of the numerator and the denominator separately and then reevaluate the limit.

Taking the derivatives: d(sin(x))/dx = cos(x) d(cos(x))/dx = -sin(x) d(cos(2x))/dx = -2sin(2x)

Now, applying L'Hôpital's Rule:

lim (x->π/4) (sin(x) - cos(x)) / cos(2x) = lim (x->π/4) (cos(x) - (-sin(x))) / (-2sin(2x))

= lim (x->π/4) (cos(x) + sin(x)) / (2sin(2x))

= (cos(π/4) + sin(π/4)) / (2sin(2 * π/4))

= (√2/2 + √2/2) / (2sin(π/2))

= (√2/2 + √2/2) / 2

= √2 / 2

Therefore, the limit of (sinx - cosx) / cos(2x) as x approaches π/4 is √2 / 2.