lim _(x → 2) (3ˣ + 3³⁻ˣ - 12)/(3^(- x/2) - 3¹⁻ˣ) is equal to_______.

Solution

Maths sir · 4 steps

Put y = 3ˣ and both top and bottom turn into simple algebra in y. Then factor and cancel.

1. Check the form. At x = 2 the numerator is 9 + 3 - 12 = 0 and the denominator is 1/3 - 1/3 = 0. So it is a 0/0 form, and x → 2 means y = 3ˣ → 9.
2. Rewrite in y. The numerator is y + 27/y - 12 = (y² - 12y + 27)/y = (y-3)(y-9)/y. The denominator is 1/√y - 3/y = (√y - 3)/y.

   (y-3)(y-9)/(√y-3) = (y-3)(√y-3)(√y+3)/(√y-3) = (y-3)(√y+3)

3. Cancel the zero. The factor y cancels between top and bottom. Writing y - 9 = (√y-3)(√y+3) removes the factor √y - 3 that made the expression 0/0.
4. So. Put y = 9: (9 - 3)(3 + 3) = 6 × 6 = 36.

The idea to keep. When a limit mixes 3ˣ, 3⁻ˣ and 3^(-x/2), substitute one power and clear denominators. The 0/0 factor then shows up and cancels, often as a difference of squares.

Answer from the NTA final answer key. Solution by PrepWiser.