Problem 47 - Calculus

Question

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

Step-by-Step Solution

Answer

The limit is \(-1\).

Step 1: Simplify the Fraction

Start by focusing on the fraction within the limit expression \( \frac{1}{1+h} - 1 \). This can be rewritten by getting a common denominator, which is \(1+h\). Thus, \(1 = \frac{1+h}{1+h} \). Now, rewrite the expression as \( \frac{1}{1+h} - \frac{1+h}{1+h} \).

Step 2: Combine the Terms

Combine the fractions from Step 1: \( \frac{1 - (1+h)}{1+h} = \frac{1 - 1 - h}{1+h} = \frac{-h}{1+h} \). This is the simplified expression for the numerator of our main limit problem.

Step 3: Substitute and Simplify the Limit Expression

The given limit expression is \( \lim _{h \rightarrow 0} \frac{1}{1+h} - \frac{1}{h} \). Substitute the simplified version of the numerator into the limit: \( \lim _{h \rightarrow 0} \frac{-h}{h(1+h)} \).

Step 4: Simplify the Fraction

Cancel \(h\) in the numerator and the denominator: \( \lim _{h \rightarrow 0} \frac{-1}{1+h} \).

Step 5: Evaluate the Limit

Now, substitute \( h = 0 \) into the simplified expression: \( \frac{-1}{1+0} = -1 \).

Key Concepts

Algebraic Manipulation

Algebraic manipulation is a powerful tool that helps us transform mathematical expressions into a simpler form, often making it easier to solve problems like evaluating limits. When faced with a limit involving complex fractions, the first step is often to simplify these fractions using algebra.

For instance, consider the expression \( \frac{1}{1+h} - 1 \). Initially, these fractions have different denominators, making them cumbersome to handle as they are.
To simplify, we find a common denominator. In this case, the common denominator is \( 1+h \), allowing us to rewrite \( 1 \) as \( \frac{1+h}{1+h} \).
This gives us:

• \( \frac{1}{1+h} - \frac{1+h}{1+h} \)

By combining these into a single fraction, we get \( \frac{1- (1+h)}{1+h} \), which simplifies further to \( \frac{-h}{1+h} \).

Algebraic manipulation like this often helps in breaking down and understanding more complex expressions.

Limit Laws

Limit laws are essential principles in calculus that allow us to evaluate limits systematically and accurately. They provide rules that handle the limits of sums, differences, products, and quotients of functions.

Consider the previously manipulated expression \( \lim_{h \to 0} \frac{-h}{h(1+h)} \). To simplify, we apply limit laws to both restructure and reduce the expression for easy evaluation.

Here's how:

• First, the -h in the numerator and h in the denominator cancel out, resulting in \( \lim_{h \to 0} \frac{-1}{1+h} \).

Now, the key limit law here is the direct substitution property, allowing us to substitute \( h \) with \( 0 \) in \( \frac{-1}{1+h} \) to find the limit value, turning the expression into \( \frac{-1}{1+0} = -1 \).

This simplification and application of limit laws highlights how these principles help us in reaching the final result efficiently.

Simplifying Expressions

Simplifying expressions is crucial in solving calculus problems, particularly in limit evaluations. Simplification involves breaking down complex expressions into more manageable parts and canceling out terms when possible.

For the limit problem \( \lim_{h \to 0} \frac{1/(1+h)-1}{h} \), simplification helped in several steps.

By identifying and canceling terms adequately:

• We simplified \( \frac{-h}{h(1+h)} \) into \( \frac{-1}{1+h} \).

This process strips down the expression to its core components, where each unnecessary term is canceled, leading to a clearer and simpler form ready for limit evaluation.

In practice, always look for factors common to both the numerator and denominator, which can be canceled to simplify the expression. This will often lead to a quicker and more straightforward solution.
These simplifications play a vital role in correctly applying other mathematical principles, such as limit laws, ensuring the solution path is both correct and intuitive.