Advertisements
Advertisements
Question
Prove that `root(3)(x^3 + 6) - root(3)(x^3 + 3)` is approximately equal to `1/x^2` when x is sufficiently large
Solution
When x is larger then `1/x` will be small (i.e.) `|1/x| < 1`
So `root(3)(x^3 + 6) - root(3)(x^3 + 3)`
= `(x^3 + 6)^(1/3) - (x^3 + 3^(1/3))`
= `{x^3(1 + 6/x^3)}^(1/3) - {x^3(1 + 3/x^3)}^(1/3)`
= `x(1 + 6/x^3)^(1/3) - x(1 + 3/x^3)^(1/3)`
= `x[1 + 1/3*6/x^3 ...] - x[ + 1/3*3/x^3 ...]`
= `(x + 2/x^2) - (x + 1/x^2)`
= `x + 2/x^2 .... - x - 1/x^2 ....`
= `1/x^2` ......(approximately)
APPEARS IN
RELATED QUESTIONS
Expand the following in ascending powers of x and find the condition on x for which the binomial expansion is valid
`1/(5 + x)`
Expand the following in ascending powers of x and find the condition on x for which the binomial expansion is valid
`2/(3 + 4x)^2`
Expand the following in ascending powers of x and find the condition on x for which the binomial expansion is valid
`(5 + x^2)^(2/3)`
Find `root(3)(10001)` approximately (two decimal places
Write the first 6 terms of the exponential series
e5x
Write the first 6 terms of the exponential series
`"e"^(-2x)`
Write the first 4 terms of the logarithmic series
`log((1 + 3x)/(1 -3x))` Find the intervals on which the expansions are valid.
Write the first 4 terms of the logarithmic series
`log((1 - 2x)/(1 + 2x))` Find the intervals on which the expansions are valid.
If y = `x + x^2/2 + x^3/3 + x^4/4 ...`, then show that x = `y - y^2/(2!) + y^3/(3!) - y^4/(4) + ...`
Find the coefficient of x4 in the expansion `(3 - 4x + x^2)/"e"^(2x)`
Find the value of `sum_("n" = 1)^oo 1/(2"n" - 1) (1/(9^("n" - 1)) + 1/(9^(2"n"- 1)))`
Choose the correct alternative:
The coefficient of x6 in (2 + 2x)10 is
Choose the correct alternative:
If a is the arithmetic mean and g is the geometric mean of two numbers, then
Choose the correct alternative:
If (1 + x2)2 (1 + x)n = a0 + a1x + a2x2 + …. + xn + 4 and if a0, a1, a2 are in AP, then n is
Choose the correct alternative:
If Sn denotes the sum of n terms of an AP whose common difference is d, the value of Sn − 2Sn−1 + Sn−2 is
Choose the correct alternative:
The value of the series `1/2 + 7/4 + 13/8 + 19/16 + ...` is
Choose the correct alternative:
The sum of an infinite GP is 18. If the first term is 6, the common ratio is
Choose the correct alternative:
The value of `1 - 1/2(2/3) + 1/3(2/3)^2 1/4(2/3)^3 + ...` is
