Advertisements
Advertisements
प्रश्न
Evaluate the following limits:
`lim_(x -> 1) (root(3)(7 + x^3) - sqrt(3 + x^2))/(x - 1)`
Advertisements
उत्तर
`lim_(x -> 1) (root(3)(7 + x^3) - sqrt(3 + x^2))/(x - 1) = lim_(x -> 1) (root(3)(7 + x^2) - 2 + 2 sqrt(3 + x^2))/(x - 1)`
= `lim_(x -> 1) ((7 + x^3)^(1/3) - 2)/(x - 1) - lim_(x -> 1) ((3 + x^2)^(1/2) - 2)/(x - 1)`
= `lim_(x -> 1) ((7 + x^3)^(1/3) - (8)^(1/3))/(x^3 - 1) xx (x^3 - 1)/(x - 1) - lim_(x -> 1) ((3 + x^2)^(1/2) - (4)^(1/2))/(x^2 - 1) xx (x^2 - 1)/(x - 1)`
= `lim_(x -> 1) ((7 + x^3)^(1/3) - (8)^(1/2))/((7 + x^3) - 8) xx ((x - 1)(x^2 + x + 1))/(x - 1) - lim_(x -> 0) ((3 + x^2)^(1/2) - (4)^(1/2))/((3 + x^2) - 4) xx ((x + 1)(x - 1))/(x - 1)`
= `lim_(x -> 1) ((7 + x^3)^(1/3) - (8)^(1/2))/((7 + x^3) - 8) xx (x^2 + x + 1) - lim_(x _> 1) ((3 + x^2)^(4)^(1/2))/((3 + x^2) - 4) xx (x + 1)`
`lim_(x -> "a") (x^"n" - "a"^"n") = "na"^("n" - 1)`
= `1/3(8)^(1/3 - 1) (1^2 + 1 + 1) - 1/2(4)^(1/2 - 1) (1 + 1)`
= `1/3(8)^(-2/3) (3) - 1/2 xx (4)^(-1/2) xx (2)`
= `(2^3)^(-2/3) - (2^2)^(- 1/2)`
= `2^(-2) - 2^(-1)`
= `1/2^2 - 1/2`
= `1/4 - 1/2`
= `(1 - 2)/4`
= `- 1/4`
APPEARS IN
संबंधित प्रश्न
Evaluate the following limit:
`lim_(y -> -3) [(y^5 + 243)/(y^3 + 27)]`
Evaluate the following limit:
`lim_(x -> 5)[(x^3 - 125)/(x^5 - 3125)]`
Evaluate the following limit :
`lim_(x -> 0)[((1 - x)^8 - 1)/((1 - x)^2 - 1)]`
Evaluate the following limit :
`lim_(x -> 7) [(x^3 - 343)/(sqrt(x) - sqrt(7))]`
In the following example, given ∈ > 0, find a δ > 0 such that whenever, |x – a| < δ, we must have |f(x) – l| < ∈.
`lim_(x -> 2) (x^2 - 1)` = 3
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 |
In exercise problems 7 – 15, use the graph to find the limits (if it exists). If the limit does not exist, explain why?
`lim_(x -> 3) (4 - x)`
Sketch the graph of f, then identify the values of x0 for which `lim_(x -> x_0)` f(x) exists.
f(x) = `{{:(sin x",", x < 0),(1 - cos x",", 0 ≤ x ≤ pi),(cos x",", x > pi):}`
Evaluate the following limits:
`lim_(x -> 3) (x^2 - 9)/(x^2(x^2 - 6x + 9))`
Evaluate the following limits:
`lim_(x -> oo) (1 + x - 3x^3)/(1 + x^2 +3x^3)`
Evaluate the following limits:
`lim_(x -> oo)(1 + 1/x)^(7x)`
Evaluate the following limits:
`lim_(x -> oo)(1 + "k"/x)^("m"/x)`
Evaluate the following limits:
`lim_(x -> oo) ((2x^2 + 3)/(2x^2 + 5))^(8x^2 + 3)`
Evaluate the following limits:
`lim_(x -> 0) (tan 2x)/x`
Evaluate the following limits:
`lim_(x -> oo) x [3^(1/x) + 1 - cos(1/x) - "e"^(1/x)]`
Choose the correct alternative:
`lim_(x - oo) sqrt(x^2 - 1)/(2x + 1)` =
Choose the correct alternative:
If `f(x) = x(- 1)^([1/x])`, x ≤ 0, then the value of `lim_(x -> 0) f(x)` is equal to
Choose the correct alternative:
`lim_(x -> 0) ("e"^(sin x) - 1)/x` =
`lim_(x→-1) (x^3 - 2x - 1)/(x^5 - 2x - 1)` = ______.
