Advertisements
Advertisements
प्रश्न
Evaluate the following limits:
`lim_(x -> 1) (sqrt(x) - x^2)/(1 - sqrt(x))`
Advertisements
उत्तर
`lim_(x -> 1) (sqrt(x) - x^2)/(1 - sqrt(x)) = lim_(x -> 1) (sqrt(x) - (sqrt(x)^4))/(1 - sqrt(x)`
= `lim_(x -> 1) (sqrt(x) (1 - (sqrt(x))^3))/(1 - sqrt(x))`
= `lim_(x -> 1) sqrt(x) (((sqrt(x))^3 - 1))/(sqrt(x) - 1)`
Put y = `sqrt(x)`
Where x → 1
We have y → `sqrt(1)` = 1
`lim_(x -> 1) (sqrt(x) - x^2)/(1 - sqrt(x)) = lim_(y -> 1) ((y^3 - 1))/(y - 1)`
= `(lim_(y -> 1) y) (lim_(y -> 1) (y^3 - 1^3)/(y - 1))`
= `1 xx 3(1)^(3 - 1)`
`lim_(x -> 1) (sqrt(x) - x^2)/(1 - sqrt(x))` = 3
APPEARS IN
संबंधित प्रश्न
Evaluate the following limit :
`lim_(x -> 7)[((root(3)(x) - root(3)(7))(root(3)(x) + root(3)(7)))/(x - 7)]`
Evaluate the following :
`lim_(x -> 0)[x/(|x| + x^2)]`
Evaluate the following :
`lim_(x -> 1) [(x + 3x^2 + 5x^3 + ... + (2"n" - 1)x^"n" - "n"^2)/(x - 1)]`
In problems 1 – 6, using the table estimate the value of the limit
`lim_(x -> 2) (x - 2)/(x^2 - 4)`
| x | 1.9 | 1.99 | 1.999 | 2.001 | 2.01 | 2.1 |
| f(x) | 0.25641 | 0.25062 | 0.250062 | 0.24993 | 0.24937 | 0.24390 |
Write a brief description of the meaning of the notation `lim_(x -> 8) f(x)` = 25
Evaluate the following limits:
`lim_(x -> 2) (2 - sqrt(x + 2))/(root(3)(2) - root(3)(4 - x))`
Evaluate the following limits:
`lim_(x - 0) (sqrt(1 + x^2) - 1)/x`
Evaluate the following limits:
`lim_(x -> 0) (sqrt(1 - x) - 1)/x^2`
Evaluate the following limits:
`lim_(x -> 5) (sqrt(x - 1) - 2)/(x - 5)`
Find the left and right limits of f(x) = `(x^2 - 4)/((x^2 + 4x+ 4)(x + 3))` at x = – 2
Evaluate the following limits:
`lim_(x -> 0)(1 + x)^(1/(3x))`
Evaluate the following limits:
`lim_(x -> oo)(1 + "k"/x)^("m"/x)`
Evaluate the following limits:
`lim_(x -> pi) (1 + sinx)^(2"cosec"x)`
Evaluate the following limits:
`lim_(x -> 0) ("e"^x - "e"^(-x))/sinx`
Evaluate the following limits:
`lim_(x -> ) (sinx(1 - cosx))/x^3`
Choose the correct alternative:
`lim_(x -> 0) (x"e"^x - sin x)/x` is
Choose the correct alternative:
If `lim_(x -> 0) (sin "p"x)/(tan 3x)` = 4, then the value of p is
Choose the correct alternative:
`lim_(x -> 0) ("e"^(sin x) - 1)/x` =
Choose the correct alternative:
The value of `lim_(x -> 0) sinx/sqrt(x^2)` is
The value of `lim_(x rightarrow 0) (sqrt((1 + x^2)) - sqrt(1 - x^2))/x^2` is ______.
