Problem 14 - Precalculus: Mathematical for Calculus

Question

Evaluate the limit, if it exists. $$\lim _{h \rightarrow 0} \frac{\sqrt{1+h}-1}{h}$$

Step-by-Step Solution

Answer

The limit is \( \frac{1}{2} \).

Step 1: Identify the form

Evaluate the expression \( \lim_{h \to 0} \frac{\sqrt{1+h} - 1}{h} \). The limit appears to have the indeterminate form \( \frac{0}{0} \) since both the numerator and denominator approach zero as \( h \) approaches zero.

Step 2: Rationalize the numerator

To resolve the indeterminate form, multiply the numerator and denominator by the conjugate of the numerator, \( \sqrt{1+h} + 1 \). This process is known as rationalizing the numerator. The expression becomes: \[ \lim_{h \to 0} \frac{(\sqrt{1+h} - 1)(\sqrt{1+h} + 1)}{h(\sqrt{1+h} + 1)} \]

Step 3: Simplify the numerator

Use the difference of squares formula \((a-b)(a+b) = a^2 - b^2\). Apply it to the numerator:\[ (\sqrt{1+h})^2 - 1^2 = 1 + h - 1 = h \].Thus, the expression simplifies to: \[ \lim_{h \to 0} \frac{h}{h(\sqrt{1+h} + 1)} \]

Step 4: Cancel common factors

Cancel the common factor of \( h \) from the numerator and the denominator:\[ \lim_{h \to 0} \frac{1}{\sqrt{1+h} + 1} \]

Step 5: Evaluate the limit

Now, substitute \( h = 0 \) into the simplified expression:\[ \frac{1}{\sqrt{1+0} + 1} = \frac{1}{1 + 1} = \frac{1}{2} \].Thus, the limit is \( \frac{1}{2} \).

Key Concepts

Indeterminate Forms

In calculus, indeterminate forms often occur when calculating limits, especially in expressions like \( \frac{0}{0} \). It means that direct substitution gives an undefined result. Identifying an indeterminate form is crucial to finding limits correctly because it signals that further steps are needed to resolve the expression.

• Indeterminate Form \( \frac{0}{0} \): Occurs when both the numerator and denominator approach zero.
• Other common indeterminate forms include \( \frac{\infty}{\infty}, 0 \cdot \infty, \infty - \infty, 0^0, \infty^0, \text{and} 1^\infty \).

Encountering an indeterminate form means you need to use algebraic techniques to simplify and find the limit. Often, these involve factorization or rationalization.

Rationalizing the Numerator

Rationalizing the numerator is a technique used to eliminate the indeterminate form in limits involving roots or radicals. By multiplying by the conjugate, you can simplify the expression to a solvable form. This process is essential when dealing with roots in numerators.

• Multiply both the numerator and the denominator by the conjugate of the numerator. The conjugate of \( \sqrt{1+h} - 1 \) is \( \sqrt{1+h} + 1 \).
• The expression becomes more manageable by converting the numerator into a difference of squares formation.

This method helps in simplifying the problem by removing the root, making the expression easier to evaluate. After simplification, terms that lead to the indeterminate form often cancel out, allowing you to find the limit.

Difference of Squares

The difference of squares formula is \( (a-b)(a+b) = a^2 - b^2 \). It's a powerful algebraic tool when simplifying expressions by eliminating indeterminate forms. In rationalizing, it's common to rewrite the expression using this formula.

• In the problem, multiplying \( (\sqrt{1+h} - 1) \) by \( (\sqrt{1+h} + 1) \) results in \( (\sqrt{1+h})^2 - 1^2 \).
• This simplifies to \( 1+h - 1 \), which equals \( h \).

Using the difference of squares formula effectively reduces the expression by transforming a complex radical into a simpler form. Once simplified, terms that appeared to cause the \( \frac{0}{0} \) form can often be canceled, allowing you to compute the limit.