Advertisements
Advertisements
प्रश्न
Find the left and right limits of f(x) = `(x^2 - 4)/((x^2 + 4x+ 4)(x + 3))` at x = – 2
Advertisements
उत्तर
f(x) = `(x^2 - 4)/((x^2 + 4x+ 4)(x + 3))` at x = – 2
f(x) = `((x + 2)(x - 2))/((x + 2)^2 (x + 3))`
f(x) = `(x - 2)/((x +2)(x +3))`
o find the let imit of f(x) at x = – 2
Put x = – 2 – h
Where h > 0
When x → – 2
We have h → 0
`lim_(x -> - 2^-) f(x) = lim_("h" -> 0) ((-2 - "h")- 2)/((-2 "h" + 2)(- 2 - "h" + 3)`
= `lim_("h" -> 0) (-4 - "h")/((- "h")(1 - "h"))`
=`lim_("h" -> 0) 1/"h" ((4 + "h")/(1 - "h"))`
= `1/0 ((4 + 0)/(1 - 0))`
= `oo`
`lim_(x -> - 2^-) f(x) = oo`
o find the right limit of f(x) at x = – 2
Put x = – 2 + h
Where h > 0
When x → – 2
We have h → 0
`lim_(x -> - 2) f(x) = lim_("h" -> 0) ((-2 + "h") - 2)/((-2 + "h" + 2)(-2 + "h" + 3))`
= `lim_("h" -> 0) (-4 + "h")/("h"(1 + "h"))`
= `lim_("h" -> 0) 1/"h"(("h" - 4)/(1 +"h"))`
= `1/0 ((0 - 4)/(1 + 0))`
= `- oo`
`lim_(x -> - 2^-) f(x) = - oo`
APPEARS IN
संबंधित प्रश्न
Evaluate the following limit:
`lim_(x -> 5)[(x^3 - 125)/(x^5 - 3125)]`
Evaluate the following limit:
If `lim_(x -> 1)[(x^4 - 1)/(x - 1)]` = `lim_(x -> "a")[(x^3 - "a"^3)/(x - "a")]`, find all possible values of a
Evaluate the following limit :
`lim_(x -> 7) [(x^3 - 343)/(sqrt(x) - sqrt(7))]`
Evaluate the following limit :
`lim_(x -> 1) [(x + x^3 + x^5 + ... + x^(2"n" - 1) - "n")/(x - 1)]`
Evaluate the following :
`lim_(x -> 0)[x/(|x| + x^2)]`
Evaluate the following :
`lim_(x -> 0) [(sqrt(1 - cosx))/x]`
In problems 1 – 6, using the table estimate the value of the limit.
`lim_(x -> 2) (x - 2)/(x^2 - x - 2)`
| x | 1.9 | 1.99 | 1.999 | 2.001 | 2.01 | 2.1 |
| f(x) | 0.344820 | 0.33444 | 0.33344 | 0.333222 | 0.33222 | 0.332258 |
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):}`
Sketch the graph of a function f that satisfies the given value:
f(– 2) = 0
f(2) = 0
`lim_(x -> 2) f(x)` = 0
`lim_(x -> 2) f(x)` does not exist.
Evaluate the following limits:
`lim_(x -> oo) (x^4 - 5x)/(x^2 - 3x + 1)`
Evaluate the following limits:
`lim_(x ->oo) (x^3/(2x^2 - 1) - x^2/(2x + 1))`
Evaluate the following limits:
`lim_(x -> oo) ((2x^2 + 3)/(2x^2 + 5))^(8x^2 + 3)`
Evaluate the following limits:
`lim_(x -> oo) (1 + 3/x)^(x + 2)`
Evaluate the following limits:
`lim_(x -> 0) (sinalphax)/(sinbetax)`
Evaluate the following limits:
`lim_(x -> 0) (tan 2x)/x`
Evaluate the following limits:
`lim_(x -> oo) ((x^2 - 2x + 1)/(x^2 -4x + 2))^x`
Evaluate the following limits:
`lim_(x -> 0) ("e"^x - "e"^(-x))/sinx`
Choose the correct alternative:
`lim_(theta -> 0) (sinsqrt(theta))/(sqrt(sin theta)`
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
The value of `lim_(x rightarrow 0) (sqrt((1 + x^2)) - sqrt(1 - x^2))/x^2` is ______.
