Intuitive Approach to the derivative of y=sin(x)
Key Questions

Well, the derivative of a function is defined using a limit, so if you are finding derivatives, then you are indeed using limits directly or indirectly; however. in most calculus classes, the derivatives of trigonometric functions are remembered as formulas once derived. So, we have the formula
#(sinx)'=cosx# .
I hope that this was helpful.
Questions
Differentiating Trigonometric Functions

Limits Involving Trigonometric Functions

Intuitive Approach to the derivative of y=sin(x)

Derivative Rules for y=cos(x) and y=tan(x)

Differentiating sin(x) from First Principles

Special Limits Involving sin(x), x, and tan(x)

Graphical Relationship Between sin(x), x, and tan(x), using Radian Measure

Derivatives of y=sec(x), y=cot(x), y= csc(x)

Differentiating Inverse Trigonometric Functions