lim x→0 of [tan(3x^2) + sin^2(5x)] / (x^2)

recall that lim x->0 of sinx/x and tanx/x = 1

You have

tan(3x^2)/x^2 + sin^2(5x)/x^2
= 3*tan(3x^2)/(3x^2) + 25(sin5x/5x)^2
--> 3+25
= 28

Hey, I did the other one, try series approach on this one too.

by the way tangent of small angle approaches sine of that small angle :) theta + theta^2/2 .......

To find the limit as x approaches 0 of the given expression:

lim x→0 [tan(3x^2) + sin^2(5x)] / (x^2)

We can start by simplifying the expression by applying basic trigonometric identities.

Let's break it down step by step:

1. First, let's simplify sin^2(5x). Since sin^2(x) = 1 - cos^2(x), we have:

sin^2(5x) = 1 - cos^2(5x)

2. Next, we can substitute this simplified expression back into our original expression:

lim x→0 [tan(3x^2) + (1 - cos^2(5x))] / (x^2)

3. Now, let's simplify the expression further. We need to simplify the numerator and denominator separately:

a. For the numerator, tan(3x^2) is a trigonometric function that tends to 0 as x approaches 0. So, we can rewrite the expression as:

lim x→0 (tan(3x^2) + 1 - cos^2(5x)) / (x^2)

Considering that the limit of tan(3x^2) as x approaches 0 is 0, we have:

lim x→0 (0 + 1 - cos^2(5x)) / (x^2)
= lim x→0 (1 - cos^2(5x)) / (x^2)

b. For the denominator, x^2 obviously approaches 0 as x approaches 0.

So, our expression becomes:

lim x→0 (1 - cos^2(5x)) / (x^2)

4. Next, let's simplify further. Notice that cos^2(5x) can be expressed as 1 - sin^2(5x):

lim x→0 (1 - (1 - sin^2(5x))) / (x^2)
= lim x→0 (sin^2(5x)) / (x^2)

5. Now, we can apply the limit properties. Since sin^2(5x) approaches 0 as x approaches 0, we are left with:

lim x→0 (0) / (x^2) = 0

Therefore, the limit as x approaches 0 of the given expression is 0.