Problem to solve
Compute \[ \lim_{ x \to \infty } [\sqrt{x^2-x}-x] \]
Solution
STEP 1
Find the following limit: lim_(x->∞)(sqrt(x^2 - x) - x)
STEP 2
Hint: | Rationalize the expression. sqrt(x^2 - x) - x = ((sqrt(x^2 - x) - x) (x + sqrt(x^2 - x)))/(x + sqrt(x^2 - x)) = -x/(x + sqrt(x^2 - x)): lim_(x->∞)(-x/(x + sqrt(x^2 - x)))
STEP 3
Hint: | Factor a constant multiple out of the limit. lim_(x->∞)-x/(x + sqrt(x^2 - x)) = -lim_(x->∞) x/(x + sqrt(x^2 - x)): -1 lim_(x->∞) x/(x + sqrt(x^2 - x))
STEP 4
Hint: | Factor 1/x out of the denominator. x/(x + sqrt(x^2 - x)) = 1/(sqrt(x^2 - x)/x + 1): -lim_(x->∞) x/(x (sqrt(x^2 - x)/x + 1))
STEP 5
Hint: | Simplify the expression inside the limit. x/(x (sqrt(x^2 - x)/x + 1)) = 1/(sqrt(x^2 - x)/x + 1): -lim_(x->∞) 1/(sqrt(x^2 - x)/x + 1)
STEP 6
Hint: | The limit of a reciprocal is the reciprocal of the limit. Using the reciprocal rule, write lim_(x->∞) 1/(sqrt(x^2 - x)/x + 1) as 1/(lim_(x->∞)(sqrt(x^2 - x)/x + 1)): -(1/(lim_(x->∞)(sqrt(x^2 - x)/x + 1)))
STEP 7
Hint: | 1 is constant, so we may pull it out of the limit. lim_(x->∞)(sqrt(x^2 - x)/x + 1) = lim_(x->∞) sqrt(x^2 - x)/x + 1: (-1)/(lim_(x->∞) sqrt(x^2 - x)/x + 1)
STEP 8
Hint: | Factor a square root out of the quotient. sqrt(x^2 - x)/x = sqrt((x^2 - x)/x^2): (-1)/(lim_(x->∞)(sqrt((x^2 - x)/x^2)) + 1)
STEP 9
Hint: | Using the power rule, pass the limit through the 2nd root. lim_(x->∞) sqrt((x^2 - x)/x^2) = sqrt(lim_(x->∞) (x^2 - x)/x^2): (-1)/(sqrt(lim_(x->∞) (x^2 - x)/x^2) + 1)
STEP 10
Hint: | Divide numerator and denominator of (x^2 - x)/x^2 by the denominator's leading term. The leading term in the denominator of (x^2 - x)/x^2 is x^2. Divide the numerator and denominator by this: (-1)/(sqrt(lim_(x->∞) (1 - 1/x)/1) + 1)
STEP 11
Hint: | Evaluate limits that tend to zero. The expression -1/x tends to zero as x approaches ∞: (-1)/(sqrt(1) + 1)
STEP 12
Hint: | Evaluate -1/(sqrt(1) + 1). -1/(sqrt(1) + 1) = -1/2: Answer: | | -1/2
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Edited by tododemate® - 2021