Problem to solve
Compute \[ \lim_{ x \to 3 } \frac{1}{[x-3]^2} \]
Solution
STEP 1
Find the following limit: lim_(x->3) 1/(x - 3)^2
STEP 2
Hint: | Apply an exponential of a logarithm to the expression. lim_(x->3) 1/(x - 3)^2 = lim_(x->3) exp(log(1/(x - 3)^2)): lim_(x->3) exp(log(1/(x - 3)^2))
STEP 3
Hint: | Apply the log power rule (log(a^b) = b log(a)). exp(log(1/(x - 3)^2)) = exp((-2 log(x - 3))): lim_(x->3) exp((-2 log(x - 3)))
STEP 4
Hint: | Pass the limit through the exponential. lim_(x->3) exp(-2 log(x - 3)) = exp(lim_(x->3)-2 log(x - 3)): (exp(lim_(x->3)-2 log(x - 3)))
STEP 5
Hint: | If neither of the terms of a product approach 0 as x->a, then by the product rule, the limit of the product is the product of the limits. Applying the product rule, write lim_(x->3)-2 log(x - 3) as -2 lim_(x->3) log(x - 3): exp(-2 lim_(x->3) log(x - 3))
STEP 6
Hint: | Pass the limit through the logarithm. lim_(x->3) log(x - 3) = log(lim_(x->3)(x - 3)): exp(-2 log(lim_(x->3)(x - 3)))
STEP 7
Hint: | The limit of a continuous function at a point is just its value there. lim_(x->3)(x - 3) = 3 - 3 = 0: exp(-2 log(0))
STEP 8
Hint: | Evaluate exp(-2 log(0)). exp(-2 log(0)) = ∞: Answer: | | ∞
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Edited by tododemate® - 2021