Evaluating Limits Analytically by Factoring Trinomials and Binomials

Factoring Trinomials with Exponents

• 💡 The problem involves evaluating a limit where the numerator is a trinomial (three-term expression) with a leading coefficient of one.
• 🎯 To factor, we need two numbers that multiply to -100 and add to 21.
• 🔑 The numbers are 25 and -4, which factor the trinomial into (x + 25)(x - 4).
• ⚠️ Special attention is given to the exponents (4 and 2), requiring the square root of x^4 (which is x^2) for factoring, resulting in (x^2 + 25)(x^2 - 4).

Factoring Differences of Squares

• 🧩 The denominator contains a difference of two perfect squares (a^2 - b^2), which factors into (a - b)(a + b).
• 🚀 The expression (x^2 - 4) is also a difference of squares, factoring into (x - 2)(x + 2).
• 🔍 The other part of the denominator, (2x + 3)^2 - 49, is factored by identifying 'a' as (2x + 3) and 'b' as 7, leading to (2x + 3 - 7)(2x + 3 + 7).
• ✅ Simplifying these terms gives (2x - 4) and (2x + 10), which can be further factored by their GCF to 2(x - 2) and 2(x + 5).

Evaluating the Limit

• ✂️ After factoring both the numerator and denominator, the (x - 2) term can be canceled out.
• 🧮 With the cancellation, direct substitution is used by replacing x with 2.
• 📈 The numerator becomes (2^2 + 25)(2 + 2) = (4 + 25)(4) = 29 * 4.
• 📊 The denominator becomes 2 * (2 + 5) = 2 * 7 = 14.
• ➗ The final simplified expression before cancellation is (29 * 4) / (2 * 7). After canceling the 4 in the numerator with a 2 from the denominator (leaving a 2 in the numerator), the result is (29 * 2) / 7. However, the transcript indicates the 4s cancel, leaving 29/7.
• ✅ The final value of the limit is 29/7.