Advertisements
Advertisements
प्रश्न
Evaluate the following limits: `lim_(x -> 0)[((1 - x)^8 - 1)/((1 - x)^2 - 1)]`
Advertisements
उत्तर १
`lim_(x -> 0)((1 - x)^8 - 1)/((1 - x)^2 - 1)`
Put 1 – x = y
As x → 0, y → 1
∴ `lim_(x -> 0)((1 - x)^8 - 1)/((1 - x)^2 - 1) = lim_(y -> 1)(y^8 - 1^8)/(y^2 - 1^2)`
= `lim_(y -> 1)((y^8 - 1^8)/(y - 1))/((y^2 - 1^2)/(y - 1)) ...[(because y -> 1"," therefore y ≠ 1","),(therefore y - 1 ≠ 0)]`
= `(lim_(y -> 1)(y^8 - 1^8)/(y - 1))/(lim_(y -> 1)(y^2 - 1^2)/(y - 1))`
= `(8(1)^7)/(2(1)^1) ...[because lim_(x -> "a")(x^"n" - "a"^"n")/(x - "a") = "na"^("n" - 1)]`
= 4
उत्तर २
`lim_(x -> 0) ((1 - x)^8 - 1)/((1 - x)^2 - 1)`
Put 1 – x = y
As x → 0, y → 1
∴ `lim_(x -> 0)((1 - x)^8 - 1)/((1 - x)^2 - 1) = lim_(y -> 1)(y^8 - 1)/(y^2 - 1)`
= `lim_(y -> 1) ((y^4 - 1)(y^4 + 1))/(y^2 - 1)`
= `lim_(y -> 1) ((y^2 - 1)(y^2 + 1)(y^4 + 1))/(y^2 - 1)`
= `lim_(y -> 1)(y^2 + 1) (y^4 + 1) ...[(because y -> 1 therefore y ≠ 1),(therefore y^2 ≠ 1),(therefore y^2 - 1 ≠ 0)]`
= (2) (2) = 4
APPEARS IN
संबंधित प्रश्न
\[\lim_{x \to 2} \frac{x^3 - 8}{x^2 - 4}\]
\[\lim_{x \to - 1/2} \frac{8 x^3 + 1}{2x + 1}\]
\[\lim_{x \to \sqrt{3}} \frac{x^2 - 3}{x^2 + 3 \sqrt{3}x - 12}\]
\[\lim_{x \to 1} \left( \frac{1}{x^2 + x - 2} - \frac{x}{x^3 - 1} \right)\]
\[\lim_{x \to 1} \left\{ \frac{x - 2}{x^2 - x} - \frac{1}{x^3 - 3 x^2 + 2x} \right\}\]
\[\lim_{x \to 0} \frac{1 - \cos mx}{x^2}\]
\[\lim_{x \to 0} \frac{1 - \cos 2x + \tan^2 x}{x \sin x}\]
\[\lim_{x \to 0} \frac{x \tan x}{1 - \cos 2x}\]
\[\lim_{x \to 0} \frac{x \cos x + \sin x}{x^2 + \tan x}\]
\[\lim_\theta \to 0 \frac{\sin 4\theta}{\tan 3\theta}\]
\[\lim_{x \to 0} \frac{8^x - 2^x}{x}\]
Evaluate the following limits: if `lim_(x -> 5)[(x^"k" - 5^"k")/(x - 5)]` = 500, find all possible values of k.
Evaluate the following Limit:
`lim_(x -> 0) ((1 + x)^"n" - 1)/x`
Evaluate: `lim_(x -> 1) ((1 + x)^6 - 1)/((1 + x)^2 - 1)`
Let `f(x) = {{:((k cos x)/(pi - 2x)",", "when" x ≠ pi/2),(3",", x = pi/2 "and if" f(x) = f(pi/2)):}` find the value of k.
Evaluate the following limit :
`lim_(x->5)[(x^3-125)/(x^5-3125)]`
Evaluate the following limit:
`lim_(x->7)[((root(3)(x)-root(3)(7))(root(3)(x)+root(3)(7)))/(x-7)]`
Evaluate the following limits: `lim_(x -> 3) [sqrt(x + 6)/x]`
Evaluate the following limit.
`lim_(x->5)[(x^3 -125)/(x^5 - 3125)]`
Evaluate the Following limit:
`lim_ (x -> 3) [sqrt (x + 6)/ x]`
