Problem 26 - Calculus: Early Transcendentals

Question

Evaluate the limit, if it exists. \( \displaystyle \lim_{t \to 0}\left( \frac{1}{t} - \frac{1}{t^2 + t} \right) \)

Step-by-Step Solution

Answer

The limit is 1.

Step 1: Combine Fractions

To simplify the expression, we first want to combine the two fractions into one. Both fractions have different denominators, so we'll find a common denominator. The common denominator of \( t \) and \( t^2 + t \) is \( t(t^2 + t) \). This gives us: \( \frac{t^2 + t}{t(t^2 + t)} - \frac{t}{t(t^2 + t)} \).

Step 2: Simplify the Expression

Combine the numerators: \( \frac{t^2 + t - t}{t(t^2 + t)} = \frac{t^2}{t(t^2 + t)} \). The expression simplifies to \( \frac{t}{t^2 + t} \), since \( t^2 \) and \( t \) are common factors in the numerator and will cancel out with \( t \) in the denominator.

Step 3: Further Simplify the Denominator

Factor the expression in the denominator: \( t^2 + t = t(t + 1) \). The expression simplifies to \( \frac{t}{t(t + 1)} \). The \( t \) in the numerator and denominator cancel out, leaving \( \frac{1}{t + 1} \).

Step 4: Evaluate the Limit

Now, substitute \( t = 0 \) in the simplified expression \( \frac{1}{t+1} \): \( \frac{1}{0+1} = 1 \). Hence, \( \lim_{t \to 0} \left( \frac{1}{t} - \frac{1}{t^2 + t} \right) = 1 \).

Key Concepts

Fraction Simplification

Fraction simplification is the process of reducing fractions to their simplest form. This is crucial in limit problems as it helps make calculations easier.

When dealing with complex expressions, simplify by performing operations like addition or subtraction to combine fractions into a single fraction.

If there are different denominators, this often involves finding a common denominator to merge the fractions, which we will discuss in the next section. Always simplify as much as possible because it helps in clearly seeing how the behavior of the fraction changes as you approach the limit.

Common Denominator

Having a common denominator is essential when you're looking to combine fractions, especially in expressions involving limits.

To find a common denominator, identify the least common multiple of the denominators involved. For example, in the exercise \( \frac{1}{t} - \frac{1}{t^2 + t} \), the denominators are \( t \) and \( t^2 + t \).

The common denominator here is \( t(t + 1) \), by factoring \( t^2 + t \) to \( t(t + 1) \). With this common denominator, you can now combine the fractions into a single fraction for further simplification.

• This step transforms the expression to: \( \frac{t^2 + t - t}{t(t^2 + t)} \)

Limit Evaluation

Limit evaluation involves calculating the value that an expression approaches as the variable within it approaches a specific value.

Typically, substitution is the first method attempted. If an expression is properly simplified, this becomes straightforward.

In our example, after simplification, we examined the limit \( \lim_{t \to 0} \frac{1}{t+1} \). By direct substitution of \( t = 0 \), the expression becomes \( \frac{1}{0+1} = 1 \).

Make sure your expression is fully simplified before substituting, as direct substitution in a complex expression can sometimes lead to indeterminate forms like \( \frac{0}{0} \).

Numerator and Denominator Cancellation

Cancellation between the numerator and the denominator is a valuable technique in simplifying fractions.

In mathematical expressions, look for common factors in the numerator and the denominator. Cancel these common factors to simplify the expression. This is particularly useful when evaluating limits, as it can eliminate problematic terms.

For instance, in the problem we started with \( \frac{t^2 + t - t}{t(t^2 + t)} \), which simplifies further to \( \frac{t}{t + 1} \) due to cancellation of \( t \) from both the numerator and the denominator.

• This simple step can often turn a complicated problem into a straightforward calculation.