Problem 50 - Calculus

Question

Use algebra to evaluate the limits. $$\lim _{h \rightarrow 0} \frac{1 / \sqrt{4+h}-1 / 2}{h}$$

Step-by-Step Solution

Answer

\(-\frac{1}{8}\)

Step 1: Simplify the Expression

We start by rewriting the given limit expression to make it easier to evaluate.Original Expression: \[\lim _{h \rightarrow 0} \frac{\frac{1}{\sqrt{4+h}}-\frac{1}{2}}{h}\]We can find a common denominator for the fractions in the numerator. The least common denominator is \(2\sqrt{4+h}\). Thus, we rewrite:\[\frac{1}{\sqrt{4+h}} - \frac{1}{2} = \frac{2 - \sqrt{4+h}}{2\sqrt{4+h}}\]The expression now becomes:\[\lim _{h \rightarrow 0} \frac{2 - \sqrt{4+h}}{2h\sqrt{4+h}}\]

Step 2: Rationalize the Numerator

We observe that the numerator \(2 - \sqrt{4+h}\) needs to be rationalized. To do this, multiply the numerator and denominator by the conjugate \(2 + \sqrt{4+h}\):\[\frac{(2 - \sqrt{4+h})(2 + \sqrt{4+h})}{2h\sqrt{4+h} \cdot (2 + \sqrt{4+h})}\]In the numerator, apply the difference of squares formula:\[(2 - \sqrt{4+h})(2 + \sqrt{4+h}) = 4 - (4+h) = -h\]This simplifies the expression to:\[\lim _{h \rightarrow 0} \frac{-h}{2h\sqrt{4+h}(2 + \sqrt{4+h})}\]

Step 3: Simplify the Expression Further

Cancel out \(h\) from the numerator and denominator:\[\lim _{h \rightarrow 0} \frac{-1}{2\sqrt{4+h}(2 + \sqrt{4+h})}\]

Step 4: Evaluate the Limit

Now that the limit is simplified, substitute \(h = 0\) in the expression:\[\frac{-1}{2\sqrt{4}(2 + \sqrt{4})} = \frac{-1}{2 \cdot 2(2 + 2)} = \frac{-1}{8}\]Thus, the limit evaluated as \(h\) approaches 0 is:

Key Concepts

Algebraic Simplification

Algebraic simplification is a crucial part of solving calculus problems involving limits. The goal is to make expressions easier to work with by transforming them into a simpler form. Consider the original limit expression: \[\lim _{h \rightarrow 0} \frac{\frac{1}{\sqrt{4+h}}-\frac{1}{2}}{h}\]To simplify this, we first find a common denominator for the fractions in the numerator.

• The least common denominator (LCD) for \(\frac{1}{\sqrt{4+h}}\) and \(\frac{1}{2}\) is \(2\sqrt{4+h}\).
• Rewriting the expression with this common denominator helps to condense the fractions.
• This simplification yields the expression \(\frac{2 - \sqrt{4+h}}{2h\sqrt{4+h}}\).

Simplifying expressions this way is crucial for reducing complexity and allowing subsequent steps to proceed more smoothly.

Rationalizing the Numerator

Rationalizing the numerator involves eliminating irrational numbers, such as square roots, from the numerator of a fraction. In this exercise, the numerator \(2 - \sqrt{4+h}\) contains a square root. To rationalize, we multiply both the numerator and the denominator by the conjugate of the numerator.

• The conjugate of \(2 - \sqrt{4+h}\) is \(2 + \sqrt{4+h}\).
• Multiplying these conjugates uses the principle of the difference of squares, resulting in a rational expression.

After performing this multiplication, the numerator becomes \(-h\), effectively removing the square root. This step is pivotal as it prepares the expression for simpler evaluation in the later stages of limit calculation.

Difference of Squares Formula

The difference of squares formula is a fundamental algebraic tool that simplifies expressions of the form \(a^2 - b^2\) into \((a-b)(a+b)\). This formula helps to simplify rational expressions with conjugates.

• In our problem, applying this formula to \((2 - \sqrt{4+h})(2 + \sqrt{4+h})\) yields \(4 - (4 + h)\).
• Simplifying \(4 - (4 + h)\) results in \(-h\).

This step is not just about changing form; it directly impacts solving the limit by eliminating complexities. Using the difference of squares formula efficiently condenses the expression, making cancellation and limit evaluation straightforward.

Substitution Method

The substitution method is a straightforward approach for evaluating limits once the expression is simplified. After algebraic simplification and rationalizing the numerator, we obtain a much more workable form of the expression. The substitution method involves directly substituting the value the variable approaches into the simplified expression.

• In our case, after cancellation, we substitute \(h = 0\) into \(\frac{-1}{2\sqrt{4+h}(2 + \sqrt{4+h})}\).
• This direct replacement simplifies to \(\frac{-1}{2\times2(2+2)} = \frac{-1}{8}\).

Performing substitution at the end confirms the final value of the limit, ensuring that the problem is cohesively solved with methodical simplification and precise calculation.