Q.31 The solution of
lim_x→8 ((x² – 64) / (x – 8)) is __________.

The limit lim_x→8 (x² − 64) / (x − 8) evaluates to 16 through algebraic simplification. Direct substitution yields the indeterminate form 0/0, but factoring reveals the solution.

This problem tests recognizing difference of squares and limit properties.

Step-by-Step Solution

Factor the numerator:

x² − 64 = (x − 8)(x + 8)

The expression becomes:

(x − 8)(x + 8) / (x − 8)

For x ≠ 8, cancel the common factor (x − 8):

x + 8

Thus:

lim_x→8 (x + 8) = 8 + 8 = 16

Alternative Approaches

• L’Hôpital’s Rule: Differentiate top and bottom → lim_x→8 2x = 16
• Numerical check: x = 8.1 ≈ 15.9, x = 7.9 ≈ 15.9
• Series expansion not needed

Introduction

The limit lim x→8 (x² − 64)/(x − 8) is a classic indeterminate form solved by factoring. This question appears in competitive exams like IIT JAM, testing algebraic simplification and limit evaluation skills.

The solution is 16, derived efficiently without advanced tools.

Detailed Solution

• Recognize 0/0 at x = 8
• Apply identity: x² − 64 = (x − 8)(x + 8)
• Simplify to x + 8 for x ≠ 8
• Evaluate limit → 8 + 8 = 16

Why No Options? Exam Context

This fill-in-the-blank style is typical in IIT JAM. If choices existed (A: 0, B: 8, C: 16, D: undefined), the correct answer would be 16.

• A: 0 ignores factoring
• B: 8 halves the correct value
• D: incorrect—discontinuity removable

Tips for Similar Limits

• Spot factorization patterns like a² − b²
• Check left & right limits
• Look for removable discontinuities before using heavy methods