Evaluate the following limit: limx → 2 x2 + 4x – 12/x2
JAMB Mathematics 2023 53 views

Evaluate the following limit: lim(x→2) (x² + 4x – 12)/(x² – 2x)

  1. 0
  2. 4 ✓
  3. 2
  4. 8

Explanation

Step 1: Try direct substitution
At x = 2:
Numerator: 2² + 4(2) – 12 = 4 + 8 – 12 = 0
Denominator: 2² – 2(2) = 4 – 4 = 0
Result: 0/0 (indeterminate – need to factor)

Step 2: Factor the numerator
x² + 4x – 12
Find factors of -12 that add to 4: 6 and -2
= (x + 6)(x – 2)

Step 3: Factor the denominator
x² – 2x = x(x – 2)

Step 4: Cancel common factor
[(x + 6)(x – 2)]/[x(x – 2)]
Cancel (x – 2):
= (x + 6)/x

Step 5: Apply limit
lim(x→2) (x + 6)/x
= (2 + 6)/2
= 8/2
= 4

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