Problem 59 - Thomas Calculus

Question

Show that the approximation of \(\sqrt{1+x}\) by its linearization at the origin must improve as \(x \rightarrow 0\) by showing that $$ \lim _{x \rightarrow 0} \frac{\sqrt{1+x}}{1+(x / 2)}=1 $$

Step-by-Step Solution

Answer

As \( x \rightarrow 0 \), the limit \( \lim _{x \rightarrow 0} \frac{\sqrt{1+x}}{1+(x / 2)}=1 \), confirming the approximation improves.

Step 1: Identify the Function and Linearization

The function we are considering is \( f(x) = \sqrt{1+x} \). We want to approximate this function by its linearization at the origin \( x = 0 \). The linearization of a function at a point is given by its tangent line at that point.

Step 2: Calculate the Derivative

Find the derivative of \( f(x) = \sqrt{1+x} \).\[ f'(x) = \frac{1}{2\sqrt{1+x}} \]At \( x = 0 \), the derivative is \( f'(0) = \frac{1}{2} \).

Step 3: Construct the Linear Approximation

The linear approximation of a function \( f(x) \) at a point \( x = a \) is given by:\[ L(x) = f(a) + f'(a)(x-a) \]For our function at \( x = 0 \), this becomes:\[ L(x) = f(0) + f'(0)(x - 0) = 1 + \frac{x}{2} \]

Step 4: Set Up the Limit Expression

We want to show that as \( x \to 0 \), the ratio of the function to its linear approximation approaches 1:\[ \lim_{x \to 0} \frac{\sqrt{1+x}}{1+\frac{x}{2}} = 1 \]

Step 5: Evaluate the Limit

To evaluate the limit, substitute the linear approximation and simplify:\[ \lim_{x \to 0} \frac{\sqrt{1+x}}{1+\frac{x}{2}} \]As \( x \to 0 \), both the numerator \( \sqrt{1+x} \) and the denominator \( 1 + \frac{x}{2} \) approach \( 1 \). The limit of the ratio of two expressions both approaching \( 1 \) as \( x \to 0 \) is clearly \( 1 \). Therefore,\[ \lim_{x \to 0} \frac{\sqrt{1+x}}{1+\frac{x}{2}} = 1 \]

Step 6: Conclude the Result

Since the limit equals 1, the approximation of \( \sqrt{1+x} \) by its linearization \( 1 + \frac{x}{2} \) improves as \( x \to 0 \). This is expected because linear approximations become more accurate the closer \( x \) is to the point of tangency, which in this case is the origin.

Key Concepts

Limit Evaluation

In calculus, evaluating limits is a fundamental concept used to analyze the behavior of functions as they approach a particular point. For example, you can use it to predict what happens as values get infinitesimally close. In our context, we want to understand how the approximation of the square root function, \(\sqrt{1+x}\), improves with its linear counterpart \(1 + \frac{x}{2}\) as \(x\) approaches zero.
The limit we are interested in is:

• \( \lim_{x \to 0} \frac{\sqrt{1+x}}{1+\frac{x}{2}} = 1 \)

By evaluating this limit, we can verify that as \(x\) gets closer to zero, the value of the square root function becomes almost identical to its linear approximation. This shows that \(1 + \frac{x}{2}\) is a good approximation for \(\sqrt{1+x}\) near the origin and demonstrates how the approximation improves at the limit point.

Derivative Calculation

The derivative of a function tells us its rate of change or slope at any given point. Calculating the derivative is crucial when forming the linear approximation of a function, as it provides the slope of the tangent line.
For \(f(x) = \sqrt{1+x}\), the derivative computation is as follows:

• \( f'(x) = \frac{1}{2\sqrt{1+x}} \)

At \(x = 0\), this simplifies to:

• \( f'(0) = \frac{1}{2} \)

We find that the slope of the tangent line at the origin is \(\frac{1}{2}\), which is essential for creating the linearization. Understanding derivatives allows us to precisely quantify how a function behaves and is fundamental in providing insights into the curve's direction and steepness.

Function Approximation

Function approximation allows us to substitute a complex function with a simpler one that closely resembles it near a particular point. Linear approximation is one of the simplest forms, using the concept of linearization.
For \(\sqrt{1+x}\), we find its linearization at the origin using:

• \( L(x) = f(0) + f'(0)(x-0) = 1 + \frac{x}{2} \)

This linear approximation captures the function's behavior very near \(x = 0\). It's particularly helpful when dealing with difficult functions or when conditions demand a simplified version of a function. Understanding linearization provides valuable insights into the initial behavior of the function and highlights how simplification can lead to more manageable calculations.

Tangent Line

The tangent line touches the curve of a function at a single point and has a slope equal to the derivative at that point. It represents the best straight-line approximation to the function at the vicinity of that point.
For \(\sqrt{1+x}\), the tangent line at \(x = 0\) is expressed by the linear equation:

• \( L(x) = 1 + \frac{x}{2} \)

This equation shows that the tangent line and the linearization are identical concepts in this context. It signifies the simplest form of local behavior monitoring of the function. Recognizing the significance of the tangent line aids in understanding the function's instant rate of change and serves as a concise way to approximate the function near the point of tangency.