I have a 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??
One of the ways to check these is to look a graph of it. Yes, it is right.
Yes, your answer is correct. Let's go through the steps to solve the limit in detail to confirm.
To evaluate the limit:
1. Start with the given expression: lim(x-->0) (1-cos(x))/(2x^2)
2. Multiply the numerator and denominator of the expression by (1+cos(x)) to get:
lim(x-->0) ((1-cos(x))*(1+cos(x)))/(2x^2*(1+cos(x)))
3. Simplify the numerator using the identity: (a^2 - b^2) = (a + b)(a - b)
lim(x-->0) (sin^2(x))/(2x^2*(1+cos(x)))
4. Use the limit property that lim(x-->0) (sin(x)/x) = 1 to simplify further:
lim(x-->0) (sin^2(x))/(2x^2*(1+cos(x)))
= (1)/(2*(1+cos(x))) [Using the limit property]
5. Substitute x = 0 into the expression obtained in Step 4:
(1)/(2*(1+cos(0)))
= (1)/(2*(1+1))
= 1/4
Therefore, the final answer is 1/4.
It's great to see that you used the identity for (sin(x)/x) to simplify the expression. Keep practicing and understanding the concepts, and you'll be able to solve more limit problems successfully.