Find The Limit By Rewriting The Fraction First

When confronted with the challenge of finding the limit of a fraction, rewriting the expression often serves as a powerful and elegant technique. This approach allows us to transform seemingly intractable problems into manageable ones, unveiling the underlying behavior of functions as they approach specific values. This comprehensive guide delves into the intricacies of this method, providing you with the knowledge and skills to confidently tackle a wide range of limit problems.

Why Rewrite Fractions to Find Limits?

Direct substitution, the most straightforward method for evaluating limits, frequently fails when applied to fractions. This failure typically manifests in two forms:

• Division by Zero: If direct substitution results in a zero in the denominator, the expression becomes undefined, and the limit cannot be determined through this method alone.

• Indeterminate Forms: Certain expressions, such as 0/0 or ∞/∞, are called indeterminate forms. These forms do not provide enough information to determine the limit's value, necessitating further analysis.

Rewriting the fraction addresses these issues by manipulating the expression to eliminate the problematic zero in the denominator or to resolve the indeterminate form. This manipulation often involves algebraic techniques like factoring, simplifying, rationalizing, or applying trigonometric identities.

Essential Algebraic Techniques for Rewriting Fractions

Several algebraic techniques are frequently employed to rewrite fractions when evaluating limits. Mastering these techniques is crucial for success.

1. Factoring: Factoring simplifies expressions by breaking them down into their constituent factors. This is particularly useful when dealing with polynomials.

   • Example: Consider the limit lim (x→2) (x² - 4) / (x - 2). Direct substitution yields 0/0. Factoring the numerator as (x - 2)(x + 2) allows us to rewrite the expression as lim (x→2) [(x - 2)(x + 2)] / (x - 2). Canceling the (x - 2) terms, we obtain lim (x→2) (x + 2), which evaluates to 4.

2. Simplifying: Simplifying involves reducing the fraction to its simplest form by canceling common factors or combining like terms.

   • Example: Consider the limit lim (x→1) (x² + 2x - 3) / (x - 1). Factoring the numerator gives (x - 1)(x + 3). The expression simplifies to lim (x→1) (x + 3), which evaluates to 4.

3. Rationalizing: Rationalizing eliminates radicals from the numerator or denominator by multiplying the fraction by a carefully chosen conjugate. This technique is particularly useful when dealing with square roots.

   • Example: Consider the limit lim (x→0) (√(x + 1) - 1) / x. Direct substitution yields 0/0. Multiplying the numerator and denominator by the conjugate √(x + 1) + 1, we get lim (x→0) [(√(x + 1) - 1)(√(x + 1) + 1)] / [x(√(x + 1) + 1)]. This simplifies to lim (x→0) (x + 1 - 1) / [x(√(x + 1) + 1)], which further simplifies to lim (x→0) x / [x(√(x + 1) + 1)]. Canceling the x terms, we get lim (x→0) 1 / (√(x + 1) + 1), which evaluates to 1/2.

4. Combining Fractions: When dealing with multiple fractions, combining them into a single fraction can simplify the expression.

   • Example: Consider the limit lim (h→0) [(1/(x + h)) - (1/x)] / h. Combining the fractions in the numerator, we get lim (h→0) [(x - (x + h)) / (x(x + h))] / h, which simplifies to lim (h→0) (-h / (x(x + h))) / h. This further simplifies to lim (h→0) -1 / (x(x + h)), which evaluates to -1/x².

5. Trigonometric Identities: When dealing with trigonometric functions, identities can be used to rewrite the expression into a more manageable form.

   • Example: Consider the limit lim (x→0) sin(x) / x. This is a fundamental limit that equals 1. While it doesn't involve rewriting the fraction in the traditional algebraic sense, understanding and recognizing this limit is crucial when dealing with trigonometric functions. Another example: lim (x→0) (1 - cos(x)) / x. Multiplying the numerator and denominator by (1 + cos(x)), we get lim (x→0) (1 - cos²(x)) / [x(1 + cos(x))]. Using the identity sin²(x) + cos²(x) = 1, we can rewrite this as lim (x→0) sin²(x) / [x(1 + cos(x))] = lim (x→0) [sin(x) / x] * [sin(x) / (1 + cos(x))] = 1 * (0 / 2) = 0.

A Step-by-Step Approach to Finding Limits by Rewriting Fractions

Here's a systematic approach to solving limit problems that involve rewriting fractions:

1. Direct Substitution: Always begin by attempting direct substitution. If this yields a defined value, you have found the limit.

2. Identify the Problem: If direct substitution results in division by zero or an indeterminate form, identify the source of the problem. Is it a factor in the denominator, a radical, or a combination of fractions?

3. Choose an Appropriate Technique: Select an algebraic technique (factoring, simplifying, rationalizing, combining fractions, trigonometric identities) that addresses the identified problem.

4. Rewrite the Fraction: Apply the chosen technique to rewrite the fraction. This may involve multiple steps.

5. Simplify: Simplify the rewritten fraction as much as possible. This often involves canceling common factors.

6. Direct Substitution (Again): Attempt direct substitution again. If the problem has been resolved, this will yield the limit.

7. Repeat if Necessary: If the second direct substitution still results in an indeterminate form, repeat steps 3-6 with a different technique or a more refined application of the previous technique.

Examples with Detailed Explanations

Let's illustrate these techniques with several examples:

Example 1: Factoring and Simplifying

Find the limit: lim (x→-3) (x² + x - 6) / (x + 3)

1. Direct Substitution: Substituting x = -3 gives (-3)² + (-3) - 6 / (-3 + 3) = 0/0 (Indeterminate Form)

2. Identify the Problem: The indeterminate form arises because (x + 3) is a factor in both the numerator and denominator, and it evaluates to zero at x = -3.

3. Choose an Appropriate Technique: Factoring the numerator.

4. Rewrite the Fraction: x² + x - 6 = (x + 3)(x - 2)

   The expression becomes: lim (x→-3) [(x + 3)(x - 2)] / (x + 3)

5. Simplify: Cancel the (x + 3) terms: lim (x→-3) (x - 2)

6. Direct Substitution (Again): Substituting x = -3 gives -3 - 2 = -5

   Therefore, lim (x→-3) (x² + x - 6) / (x + 3) = -5

Example 2: Rationalizing the Numerator

Find the limit: lim (x→0) (√(4 + x) - 2) / x

1. Direct Substitution: Substituting x = 0 gives (√(4 + 0) - 2) / 0 = 0/0 (Indeterminate Form)

2. Identify the Problem: The indeterminate form arises from the square root in the numerator.

3. Choose an Appropriate Technique: Rationalizing the numerator.

4. Rewrite the Fraction: Multiply the numerator and denominator by the conjugate of the numerator, which is √(4 + x) + 2:

   lim (x→0) [(√(4 + x) - 2) / x] * [(√(4 + x) + 2) / (√(4 + x) + 2)]

   This simplifies to: lim (x→0) (4 + x - 4) / [x(√(4 + x) + 2)]

   Which further simplifies to: lim (x→0) x / [x(√(4 + x) + 2)]

5. Simplify: Cancel the x terms: lim (x→0) 1 / (√(4 + x) + 2)

6. Direct Substitution (Again): Substituting x = 0 gives 1 / (√(4 + 0) + 2) = 1 / (2 + 2) = 1/4

   Therefore, lim (x→0) (√(4 + x) - 2) / x = 1/4

Example 3: Combining Fractions and Simplifying

Find the limit: lim (x→2) [1/(x - 2) - 4/(x² - 4)]

1. Direct Substitution: Substituting x = 2 gives 1/(2 - 2) - 4/(2² - 4) = 1/0 - 4/0 (Undefined)

2. Identify the Problem: Division by zero.

3. Choose an Appropriate Technique: Combining the fractions into a single fraction and simplifying.

4. Rewrite the Fraction: First, factor the denominator x² - 4 = (x - 2)(x + 2).

   Then, rewrite the expression with a common denominator:

   lim (x→2) [ (x + 2) / ((x - 2)(x + 2)) - 4 / ((x - 2)(x + 2)) ]

   Combining the fractions: lim (x→2) [ (x + 2 - 4) / ((x - 2)(x + 2)) ]

   Simplifying: lim (x→2) [ (x - 2) / ((x - 2)(x + 2)) ]

5. Simplify: Cancel the (x - 2) terms: lim (x→2) 1 / (x + 2)

6. Direct Substitution (Again): Substituting x = 2 gives 1 / (2 + 2) = 1/4

   Therefore, lim (x→2) [1/(x - 2) - 4/(x² - 4)] = 1/4

Example 4: Trigonometric Identities

Find the limit: lim (x→0) tan(x) / x

1. Direct Substitution: Substituting x = 0 gives tan(0) / 0 = 0/0 (Indeterminate Form)

2. Identify the Problem: The indeterminate form arises from the tangent function and the division by zero.

3. Choose an Appropriate Technique: Using the trigonometric identity tan(x) = sin(x) / cos(x)

4. Rewrite the Fraction: lim (x→0) [sin(x) / cos(x)] / x = lim (x→0) sin(x) / [x * cos(x)]

5. Simplify: Rewrite as a product of limits: lim (x→0) [sin(x) / x] * [1 / cos(x)]

6. Evaluate the Limits: We know that lim (x→0) sin(x) / x = 1 and lim (x→0) 1 / cos(x) = 1 / cos(0) = 1 / 1 = 1

   Therefore, lim (x→0) tan(x) / x = 1 * 1 = 1

Common Mistakes to Avoid

• Forgetting to Check for Indeterminate Forms: Always check if direct substitution results in an indeterminate form before attempting to rewrite the fraction.

• Incorrect Factoring: Ensure that factoring is done correctly. Double-check your work.

• Multiplying by the Wrong Conjugate: When rationalizing, make sure you are multiplying by the correct conjugate. The conjugate of a + b is a - b.

• Incorrectly Applying Trigonometric Identities: Use trigonometric identities correctly. Refer to a list of identities if needed.

• Not Simplifying Completely: Simplify the expression as much as possible after rewriting the fraction. This makes the subsequent direct substitution easier.

Advanced Techniques and Considerations

While the techniques described above cover a wide range of limit problems, some situations require more advanced approaches.

• L'Hôpital's Rule: L'Hôpital's Rule provides a powerful method for evaluating limits of indeterminate forms (0/0 or ∞/∞). It states that if lim (x→c) f(x) / g(x) is of the form 0/0 or ∞/∞, then lim (x→c) f(x) / g(x) = lim (x→c) f'(x) / g'(x), provided the latter limit exists. However, L'Hôpital's Rule should only be applied after confirming that the limit is indeed an indeterminate form.

• Squeeze Theorem (Sandwich Theorem): The Squeeze Theorem is useful for finding the limit of a function that is "squeezed" between two other functions whose limits are known. If g(x) ≤ f(x) ≤ h(x) for all x in an interval containing c (except possibly at c), and lim (x→c) g(x) = lim (x→c) h(x) = L, then lim (x→c) f(x) = L.

• Limits at Infinity: When dealing with limits as x approaches infinity (∞) or negative infinity (-∞), divide both the numerator and denominator by the highest power of x in the denominator. This simplifies the expression and allows you to evaluate the limit.

Practice Problems

To solidify your understanding, try solving the following practice problems:

1. lim (x→1) (x² - 1) / (x - 1)
2. lim (x→0) (√(x + 9) - 3) / x
3. lim (x→-2) (x² + 5x + 6) / (x + 2)
4. lim (x→0) (sin(2x)) / x
5. lim (x→3) [1/(x - 3) - 6/(x² - 9)]

Conclusion

Finding limits by rewriting fractions is a fundamental skill in calculus. By mastering algebraic techniques like factoring, simplifying, rationalizing, combining fractions, and applying trigonometric identities, you can effectively tackle a wide variety of limit problems. Remember to follow a systematic approach, check for indeterminate forms, and simplify your expressions thoroughly. With practice, you'll become proficient in rewriting fractions to unveil the hidden behavior of functions and confidently determine their limits. Remember to always double-check your work and consider using advanced techniques like L'Hôpital's Rule or the Squeeze Theorem when appropriate. Good luck!