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
\[\lim_{x \to 2} \left( 3 - x \right)\]
\[\lim_{x \to - 1} \frac{x^3 - 3x + 1}{x - 1}\]
\[\lim_{x \to 3} \frac{x^4 - 81}{x^2 - 9}\]
\[\lim_{x \to 5} \frac{x^3 - 125}{x^2 - 7x + 10}\]
\[\lim_{x \to 1} \frac{x^{15} - 1}{x^{10} - 1}\]
\[\lim_{x \to a} \frac{x^{2/3} - a^{2/3}}{x^{3/4} - a^{3/4}}\]
\[\lim_{n \to \infty} \frac{n^2}{1 + 2 + 3 + . . . + n}\]
\[\lim_{x \to 0} \frac{\cos 3x - \cos 7x}{x^2}\]
\[\lim_{x \to 0} \frac{\sin 5x - \sin 3x}{\sin x}\]
\[\lim_{x \to 0} \frac{\sec 5x - \sec 3x}{\sec 3x - \sec x}\]
\[\lim_{x \to \pi} \frac{\sqrt{2 + \cos x} - 1}{\left( \pi - x \right)^2}\]
Evaluate the following limit:
\[\lim_{x \to \pi} \frac{1 - \sin\frac{x}{2}}{\cos\frac{x}{2}\left( \cos\frac{x}{4} - \sin\frac{x}{4} \right)}\]
Write the value of \[\lim_{x \to 2} \frac{\left| x - 2 \right|}{x - 2} .\]
\[\lim_{x \to \infty} \frac{\sin x}{x}\] equals
The value of \[\lim_{x \to \infty} \frac{\sqrt{1 + x^4} + \left( 1 + x^2 \right)}{x^2}\] is
Which of the following function is not continuous at x = 0?
Evaluate `lim_(h -> 0) ((a + h)^2 sin (a + h) - a^2 sina)/h`
If f(x) = `{{:(1 if x "is rational"),(-1 if x "is rational"):}` is continuous on ______.
Evaluate the following limit:
`lim_(x->3)[(sqrt(x+6))/x]`
Evaluate the Following limit:
`lim_ (x -> 3) [sqrt (x + 6)/ x]`
