Limx-0 sinx/tanx+tanx.sinx

The first step in evaluating a limit would be to substitute the limiting value into the expression. If it evaluates to a valid number, that is the limit.

Example:
limit x->1 x²/x =1²/1 = 1

Since
sinx/tanx + tanx.sinx ≡ 1/cos(x)
so the limit is 1/cos(0)=1.

To evaluate the limit of the expression limx→0 sin(x)/(tan(x) + tan(x)sin(x)), we can use algebraic manipulation and trigonometric identities. Here's how you can solve it step by step:

Step 1: Simplify the expression
sin(x) can be rewritten as sin(x)/1
We can multiply the expression by 1, using the identity tan(x) = sin(x)/cos(x)
So, sin(x)/(tan(x) + tan(x)sin(x)) becomes sin(x)*(cos(x)/sin(x))/(tan(x)*(cos(x)/sin(x)) + tan(x)*sin(x)*(cos(x)/sin(x)))

Step 2: Cancel out the common factors
sin(x) cancels out with sin(x) in the numerators and denominators,
cos(x) cancels out with cos(x) in the numerators and denominators, and
tan(x) cancels out with 1/sin(x) in the denominators.

This simplifies the expression to cos(x)/(1 + cos(x)).

Step 3: Evaluate the limit
Now, we can substitute x = 0 into the simplified expression to find the limit.
limx→0 cos(x)/(1 + cos(x))

Since cosine is a continuous function, we can directly substitute x = 0 into the expression:
cos(0)/(1 + cos(0))

cos(0) = 1, and
1 + cos(0) = 1 + 1 = 2

So, the final result is 1/2.

Therefore, the value of limx→0 sin(x)/(tan(x) + tan(x)sin(x)) is 1/2.