Limits-10 - Problems with Complexity

Lim x→∞ [√(x²+1) - ³√(x³+1)]/[⁴√(x⁴+1) - ⁵√(x³+1)

a. 0
b. -1
c. 1
d. limit does not exist

Answer: (a)

When limits x→∞ are to be calculated, polynomials have to divided by their highest power term throughout. In this case, dividing numerator and denominator by x will result in the expression with in the root symbols getting divided by highest power term.

Lim x→∞ [√(x²+1) - ³√(x³+1)]/[⁴√(x⁴+1) - ⁵√(x³+1)]

= Lim x→∞ [√(1+(1/x²)) - ³√(1+(1/x³))]/[⁴√(1+(1/x⁴)) - ⁵√((1/x²+(1/x⁵))]

= (1-1/(1-0) = 0

1/x and Other terms with highest powers of x in the denominator approach zero as x→∞