How do I solve lim(x-->pi/4)(sinx-cosx)/cos(2x)
The (x-->pi/4) goes below the lim.
Should i just substitute the x's with pi/4??
This question is related to limits.
If you do that, you will get 0/0. You have to take the ratio of the derivatives.
lim(x->pi/4)(sinx-cosx)/cos(2x)
= lim(x->pi/4)(cosx+sinx)/-2 sin(2x)
= sqrt 2/(-2) = -0.707
That's called using L'Hopital's rule for indeterminate-ratio limits. It works for 0/0 or infinity/infinity. Try it with a number close to pi/4, such as 0.785, and you will see that it gives the right answer.
if you put in PI/4 in to the x, you will get an indeterminate form 0/0.
Lim (sinx/cos2x - cosx/cos2x)
multiply numerator and denominator by (sinx+cosx)
lim (sin^2x -cos^2x)/cos2x(sinx+cosx)
lim (sin^2x-cos^2x)/(sin^2x-cos^2x)(sinx+cosx)
lim 1/(sinx+cosx)= 1/sqrt2
check that.
looking at drwls answer, using L'Hopitals rule (I assumed you had not gotten to it yet), I see I must have made a sign error, you can find it.
Thanks. Actually i did not learn that other rule yet and I don't think we will but thanks anyway it was helpful.
I solved using bobpursley's idea and got sqrt2/2=1/sqrt2.
Thanks everyone!
I have a similar question relating to limits that I solved
lim(x-->0) (1-cosx)/2x^2
I multiplied the numerator and denominator by (1+cosx) to get
lim(x->0) (1-cos^2x)/2x^2(1+cosx)
= lim(x->0)sin^2x/2x^2(1+cosx)
the lim(x->0) (sinx/x)^2 would =1
I then substituted the remaining portion with 0 and got 1/4
Is this answer right??
To solve the limit: lim(x-->pi/4) (sin(x) - cos(x))/cos(2x), you can't simply substitute x with pi/4. Here's the step-by-step process to solve it:
Step 1: Simplify the expression
Begin by simplifying the expression within the limit. Rewrite cos(2x) as cos^2(x) - sin^2(x) using the identity cos(2x) = cos^2(x) - sin^2(x). The expression then becomes:
lim(x-->pi/4) (sin(x) - cos(x))/(cos^2(x) - sin^2(x))
Step 2: Factorize the denominator
Factorize the denominator using the identity a^2 - b^2 = (a + b)(a - b). The denominator becomes:
lim(x-->pi/4) (sin(x) - cos(x))/((cos(x) + sin(x))(cos(x) - sin(x)))
Step 3: Simplify further
Since cos(x) + sin(x) is a non-zero term, divide both the numerator and denominator by (cos(x) + sin(x)):
lim(x-->pi/4) (sin(x) - cos(x))/(cos(x) + sin(x))(cos(x) - sin(x))
Step 4: Apply the limit
Now, you can substitute x with pi/4:
(sin(pi/4) - cos(pi/4))/((cos(pi/4) + sin(pi/4))(cos(pi/4) - sin(pi/4)))
Step 5: Evaluate the expression
Evaluate the expression further:
[(√2/2) - (√2/2)] / [(√2/2) + (√2/2)][(√2/2) - (√2/2)]
The numerator becomes 0, and the denominator simplifies:
(1/2)(-1)(1/2) = -1/4
Therefore, the solution to the limit lim(x-->pi/4) (sin(x) - cos(x))/cos(2x) is -1/4.