109-112 {Use of Tech} Calculating limits The following limits are the derivatives of a composite function g at a point a.
b. Use the Chain Rule to find each limit. Verify your answer by using a calculator.
$\underset{x\to 0}{\lim }\frac{\sqrt{4+\sin \left(x\right)}-2}{x}$

Verified step by step guidance

1. Step 1: Recognize that the given limit is in the form of a derivative. The expression \( \lim_{x \to 0} \frac{\sqrt{4 + \sin(x)} - 2}{x} \) can be interpreted as the derivative of a composite function at a point.
2. Step 2: Identify the outer function \( f(u) = \sqrt{u} \) and the inner function \( u(x) = 4 + \sin(x) \). The point of interest is \( x = 0 \).
3. Step 3: Apply the Chain Rule for derivatives, which states that \( g'(x) = f'(u(x)) \cdot u'(x) \). First, find \( f'(u) \) by differentiating \( f(u) = \sqrt{u} \), which gives \( f'(u) = \frac{1}{2\sqrt{u}} \).
4. Step 4: Differentiate the inner function \( u(x) = 4 + \sin(x) \) to find \( u'(x) = \cos(x) \).
5. Step 5: Evaluate the derivative at \( x = 0 \). Substitute \( u(0) = 4 + \sin(0) = 4 \) into \( f'(u) \) to get \( f'(4) = \frac{1}{2\sqrt{4}} = \frac{1}{4} \). Then, multiply by \( u'(0) = \cos(0) = 1 \) to find the limit.

Key Concepts

Limits

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It helps in understanding how functions behave near points of interest, including points of discontinuity or where they are not explicitly defined. In this question, the limit as x approaches 0 is crucial for evaluating the expression involving the square root and sine function.

Chain Rule

The Chain Rule is a formula for computing the derivative of a composite function. It states that if you have a function that is the composition of two functions, the derivative can be found by multiplying the derivative of the outer function by the derivative of the inner function. In this context, applying the Chain Rule is essential for finding the derivative of the function involved in the limit.

Derivatives

A derivative represents the rate of change of a function with respect to its variable. It is a key concept in calculus that provides information about the slope of the function at any given point. In this problem, the limit being evaluated is actually the derivative of the function g at the point a, which is determined using the limit definition of the derivative.