Step 1: Understanding the Concept:
The given limit expression matches the precise limit definition of a derivative at a specific point.
\[ \lim_{x\rightarrow a} \frac{F(x) - F(a)}{x - a} = F'(a) \] Step 2: Key Formula or Approach:
Let $F(x) = f(g(h(x)))$. We need to find the derivative $F'(x)$ using the Chain Rule and evaluate it at $x = 1$.
Step 3: Detailed Explanation:
First, construct the composite function $F(x)$: \[ F(x) = f(g(x^2)) = f(\cos(x^2)) = \sin(\cos(x^2)) \] Now, differentiate $F(x)$ with respect to $x$: \[ F'(x) = \frac{d}{dx} \left[ \sin(\cos(x^2)) \right] \] Apply the Chain Rule sequentially (outside to inside): \[ F'(x) = \cos(\cos(x^2)) \cdot \frac{d}{dx}[\cos(x^2)] \] \[ F'(x) = \cos(\cos(x^2)) \cdot [-\sin(x^2)] \cdot \frac{d}{dx}[x^2] \] \[ F'(x) = \cos(\cos(x^2)) \cdot [-\sin(x^2)] \cdot (2x) \] \[ F'(x) = -2x \sin(x^2) \cos(\cos(x^2)) \] Finally, evaluate the derivative at $x = 1$: \[ F'(1) = -2(1) \sin(1^2) \cos(\cos(1^2)) \] \[ F'(1) = -2 \sin(1) \cos(\cos 1) \] Step 4: Final Answer:
The limit is $-2 \sin 1 \cos(\cos 1)$.