the limit represents the derivative of some function f at some number a. state f and a.
lim (sin(pai/2 + t)-1)/t
t->0
Since df/dx = lim(t->0) [f(x+t)-f(x)]/t
f(a+t) - f(a) = sin(pi/2 + t) - sin(pi/2)
Looks like we have f(x) = sin(x) and a = pi/2
To find the function f and the number a, we first need to simplify the given expression, and then apply the definition of the derivative.
Let's work through it step by step:
Step 1: Simplify the expression
The given expression is (sin(π/2 + t) - 1) / t. We can simplify it further by using the trigonometric identity sin(A + B) = sin(A)cos(B) + cos(A)sin(B).
(sin(π/2 + t) - 1) / t
= (sin(π/2)cos(t) + cos(π/2)sin(t) - 1) / t
= (1 * cos(t) + 0 * sin(t) - 1) / t
= (cos(t) - 1) / t
So, we have a simplified expression of (cos(t) - 1) / t.
Step 2: Apply the definition of the derivative
To find the function f and the number a, we need to rewrite the expression in the form of the derivative definition:
lim (cos(t) - 1) / t
t->0
This limit represents the derivative of the function f at the number a, where f(x) = cos(x) and a = 0.
Therefore, the function f is f(x) = cos(x), and the number a is a = 0.