1) Limit theta approaching zero

sin squared theta/ tan theta

2) Lim x appraoches zero
x + sin x/x
x + sin x/x = x/x + sinx/x
0 + 1 =1

My answer : 1 Is this correct

If you mean (x + sinx)/x, the limit is 2.

If you mean x + (sinx/x), the limit is 1

Yes, your answer is correct. Let's go through the steps to find the limit and understand why it is equal to 1.

For the first question, we are asked to find the limit of the expression (sin^2(theta)) / tan(theta) as theta approaches 0.

To find the limit, we can simplify the expression using trigonometric identities. We know that tan(theta) = sin(theta) / cos(theta).

So, the expression becomes (sin^2(theta)) / (sin(theta) / cos(theta)), which can be further simplified by multiplying the numerator and denominator by cos(theta):
= sin^2(theta) * cos(theta) / sin(theta)

Now, we simplify using the identity sin^2(theta) = 1 - cos^2(theta):
= (1 - cos^2(theta)) * cos(theta) / sin(theta)
= cos(theta) - cos^3(theta) / sin(theta)

Next, we use the fact that cos(theta) approaches 1 and sin(theta) approaches 0 as theta approaches 0. So, as theta approaches 0, the second term in the expression (cos^3(theta)) / sin(theta) will approach 0, whereas the first term (cos(theta)) will approach 1.

Therefore, the limit of the expression as theta approaches 0 is 1.

For the second question, you correctly simplified the expression x + sin(x) / x as x / x + sin(x) / x. Then, you substituted 0 for x, which gives the result 1. So your answer is correct.