Advertisements
Advertisements
Question
Evaluate the following limits: `lim_(x -> 0)[(root(3)(1 + x) - sqrt(1 + x))/x]`
Advertisements
Solution
`lim_(x -> 0)(root(3)(1 + x) - sqrt(1 + x))/x`
= `lim_(x -> 0)((1 + x)^(1/3) - (1 + x)^(1/2))/x`
Put 1 + x = y
As x → 0, y → 1
∴ `lim_(x -> 0)((1 + x)^(1/3) - (1 + x)^(1/2))/x`
= `lim_(y -> 1)(y^(1/3) - y^(1/2))/(y - 1)`
= `lim_(y -> 1)((y^(1/3) - 1) - (y^(1/2) - 1))/(y - 1)`
= `lim_(y -> 1)((y^(1/3) - 1)/(y - 1) - (y^(1/2) - 1)/(y - 1))`
= `lim_(y -> 1) (y^(1/3) - 1^(1/3))/(y - 1) - lim_(y -> 1)(y^(1/2) - 1^(1/2))/(y - 1)`
= `1/3(1)^((-2)/3) - 1/2(1)^((-1)/2) ...[because lim_(x -> "a") (x^"n" - "a"^"n")/(x - "a") = "na"^("n" - 1)]`
= `1/3 - 1/2`
= `(2 - 3)/6`
= `-1/6`
APPEARS IN
RELATED QUESTIONS
\[\lim_{x \to - 5} \frac{2 x^2 + 9x - 5}{x + 5}\]
\[\lim_{x \to 2} \frac{x^3 + 3 x^2 - 9x - 2}{x^3 - x - 6}\]
\[\lim_{x \to 1} \left\{ \frac{x - 2}{x^2 - x} - \frac{1}{x^3 - 3 x^2 + 2x} \right\}\]
\[\lim_{x \to - 1/2} \frac{8 x^3 + 1}{2x + 1}\]
\[\lim_{n \to \infty} \left[ \frac{1^3 + 2^3 + . . . n^3}{\left( n - 1 \right)^4} \right]\]
\[\lim_{x \to \infty} \left[ \frac{x^4 + 7 x^3 + 46x + a}{x^4 + 6} \right]\] where a is a non-zero real number.
\[\lim_{h \to 0} \frac{\left( a + h \right)^2 \sin \left( a + h \right) - a^2 \sin a}{h}\]
\[\lim_{x \to 0} \frac{\sin 3x + 7x}{4x + \sin 2x}\]
\[\lim_{x \to a} \frac{\sin \sqrt{x} - \sin \sqrt{a}}{x - a}\]
\[\lim_{x \to 1} \frac{1 - x^2}{\sin \pi x}\]
\[\lim_{x \to \frac{\pi}{4}} \frac{1 - \sin 2x}{1 + \cos 4x}\]
\[\lim_{x \to 1} \frac{1 - \frac{1}{x}}{\sin \pi \left( x - 1 \right)}\]
\[\lim_{x \to \frac{\pi}{4}} \frac{\sqrt{2} - \cos x - \sin x}{\left( 4x - \pi \right)^2}\]
\[\lim_{x \to \frac{\pi}{6}} \frac{\cot^2 x - 3}{cosec x - 2}\]
\[\lim_{x \to 0} \frac{\log \left( a + x \right) - \log a}{x}\]
The value of \[\lim_{x \to \infty} \frac{\sqrt{1 + x^4} + \left( 1 + x^2 \right)}{x^2}\] is
\[\lim_{x \to 3} \frac{\sum^n_{r = 1} x^r - \sum^n_{r = 1} 3^r}{x - 3}\]is real to
Evaluate the following limits: `lim_(y -> 1) [(2y - 2)/(root(3)(7 + y) - 2)]`
Evaluate the following limit :
`lim_(x->5)[(x^3-125)/(x^5-3125)]`
