\(\lim _{h \rightarrow 0} \frac{1}{h}\left[\int_{a}^{x+h} \sin ^{4} t d t-\int_{a}^{x} \sin ^{4} t d t\right]=\) (1) \(\sin ^{4} x\) (2) \(4 \sin ^{3} x \cos x\) (3) 0 (4) \(\frac{\sin ^{5} x}{5}\)

Limits help us understand the behavior of functions as they approach specific points. In the context of calculus, limits can define continuity, derivatives, and integrals.
For the given problem, the goal is to evaluate the limit expression: \ \lim _{h \rightarrow 0} \frac{1}{h}\bigg[\big( \int_{a}^{x+h} \sin ^{4} t \ dt - \int_{a}^{x} \sin ^{4} t \ dt \big) \bigg]
Initially, this expression might seem complex, but by breaking it down, it's easier to handle. The numerator here represents the difference between two definite integrals. Understanding this difference is key to evaluating the limit.
Specifically, the numerator can be viewed as the integral of \( \sin^4 t \) from \( x \) to \( x + h \). The limit then seeks to determine the average value of this function over a very small interval, which gives insight into the function's behavior at point \( x \). Understanding how to simplify and break down such expressions is fundamental in advanced calculus.

The Mean Value Theorem for integrals states that for a continuous function \( f \) over the interval \( [a, b] \), there exists a point \( c \) in \( [a, b] \) such that: \ \int_{a}^{b} f(t) \ dt = f(c) \cdot (b - a)
This theorem bridges the gap between the average value of a function and its integral over an interval.
In our problem, we apply this theorem to \( \sin^4 t \) over \( [x, x+h] \). According to the theorem, there exists a point \( c \) in this interval for which: \ \int_{x}^{x+h} \sin^4(t) \ dt = \sin^4(c) \cdot h
This representation allows us to replace the integral with a simpler product, making the limit easier to evaluate. The constant \( h \) in the product will cancel out in the fraction, simplifying the expression further. By leveraging the Mean Value Theorem for integrals, we efficiently manage the complexity of the problem.

Understanding integral properties is crucial for solving calculus problems. Some essential properties include: \begin{itemize} \item Linearity: \( \int(af(x) + bg(x)) \ dx = a \int f(x) \ dx + b \int g(x) \ dx \) \item Additivity: \( \int_{a}^{b} f(x) \ dx + \int_{b}^{c} f(x) \ dx = \int_{a}^{c} f(x) \ dx \) \item Substitution: Changing the variable of integration affects the bounds and the function itself. \end{itemize}
For our specific problem, we rely on the property that if \( F(x) \) is the antiderivative of \( f(x) \), then: \ \int_{a}^{x+h} f(t) \ dt - \int_{a}^{x} f(t) \ dt = \int_{x}^{x+h} f(t) \ dt
This allows us to focus only on the integral over \( [x, x+h] \), simplifying the original expression: \ \lim _{h \rightarrow 0} \frac{1}{h} \int_{x}^{x+h} \sin^4 t \ dt
Another vital property used is the ability to adapt our integral expression to fit the form used in the Mean Value Theorem for integrals. By understanding and applying relevant properties of integrals, we effectively analyze and simplify our given limit expression.