Advertisements
Advertisements
प्रश्न
Evaluate the following limits:
`lim_(x -> 3) (x^2 - 9)/(x^2(x^2 - 6x + 9))`
Advertisements
उत्तर
`lim_(x -> 3) (x^2 - 9)/(x^2(x^2 - 6x + 9)) = lim_(x -> 3) ((x + 3)(x - 3))/(x^2(x - 3)2`
= `lim_(x -> 3) (x +3)/(x^2(x - 3))`
To find he left limit
Put x = 3 – h
Where h > 0
When x → 3
We have h → 0
`lim_(x -> 3^-) (x^2 - 9)/(x^2(x^2 - 6x + 9)) = lim_("h" -> 0) (3 - "h" + 3)/((3 - "h")^2 (3 - "h" - 3))`
= `lim_("h" -> 0) (6 - "h")/(-"h"(3- "h")^2`
= `- lim_("h" -> 0) (6 - "h")/("h"(3 - "h")^2`
= `- (6 - 0)/(0(3 - 0)^2`
= `- 6/0`
`lim_(x -> 3^-) (x^2 - 9)/(x^2(x^2 - 6x + 9)) = - oo`
To find he right limit
Put x = 3 + h
Where h > 0
When x → 3
We have h → 0
`lim_(x -> 3^+) (x^2 - 9)/(x^2(x^2 - 6x + 9)) = lim_("h" -> 0) (3 + "h" + 3)/((3 + "h")^2 (3 + "h" - 3))`
= `lim_("h" -> 0) (6 + "h")/("h"(3 + "h")^2`
= `lim_("h" -> 0) (6 + "h")/("h"(3 + "h")^2`
= `(6 + 0)/(0(3 + 0)^2`
= `6/0`
`lim_(x -> 3^+) (x^2 - 9)/(x^2(x^2 - 6x + 9)) = oo`
APPEARS IN
संबंधित प्रश्न
In the following example, given ∈ > 0, find a δ > 0 such that whenever, |x – a| < δ, we must have |f(x) – l| < ∈.
`lim_(x -> 1) (x^2 + x + 1)` = 3
Evaluate the following :
`lim_(x -> 0) [(sqrt(1 - cosx))/x]`
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)`
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) 1/(x - 3)`
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 -> 5) |x - 5|/(x - 5)`
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 -> x/2) tan x`
Sketch the graph of a function f that satisfies the given value:
f(0) is undefined
`lim_(x -> 0) f(x)` = 4
f(2) = 6
`lim_(x -> 2) f(x)` = 3
Evaluate the following limits:
`lim_(sqrt(x) -> 3) (x^2 - 81)/(sqrt(x) - 3)`
Evaluate the following limits:
`lim_(x -> 1) (sqrt(x) - x^2)/(1 - sqrt(x))`
Find the left and right limits of f(x) = `(x^2 - 4)/((x^2 + 4x+ 4)(x + 3))` at x = – 2
Find the left and right limits of f(x) = tan x at x = `pi/2`
Evaluate the following limits:
`lim_(x -> oo) (x^4 - 5x)/(x^2 - 3x + 1)`
Evaluate the following limits:
`lim_(x -> oo)(1 + "k"/x)^("m"/x)`
Evaluate the following limits:
`lim_(x -> 0) (3^x - 1)/(sqrt(x + 1) - 1)`
Choose the correct alternative:
`lim_(x -> oo) sinx/x`
Choose the correct alternative:
`lim_(x -> 0) sqrt(1 - cos 2x)/x`
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` =
`lim_(x→∞)((x + 7)/(x + 2))^(x + 4)` is ______.
