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
Evaluate the following limit:
`lim_(x -> 3)[sqrt(2x + 6)/x]`
Evaluate the following limit :
If `lim_(x -> 5) [(x^"k" - 5^"k")/(x - 5)]` = 500, find all possible values of k.
Evaluate the following :
`lim_(x -> 0)[x/(|x| + x^2)]`
Evaluate the following :
`lim_(x -> 0) {1/x^12 [1 - cos(x^2/2) - cos(x^4/4) + cos(x^2/2) cos(x^4/4)]}`
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 -> 1) (x^2 + 2)`
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(0) is undefined
`lim_(x -> 0) f(x)` = 4
f(2) = 6
`lim_(x -> 2) f(x)` = 3
If the limit of f(x) as x approaches 2 is 4, can you conclude anything about f(2)? Explain reasoning
Evaluate the following limits:
`lim_(x -> 1) (sqrt(x) - x^2)/(1 - sqrt(x))`
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) - 1)/x^2`
Evaluate the following limits:
`lim_(x -> oo) (1 + x - 3x^3)/(1 + x^2 +3x^3)`
Show that `lim_("n" -> oo) (1 + 2 + 3 + ... + "n")/(3"n"^2 + 7n" + 2) = 1/6`
A tank contains 5000 litres of pure water. Brine (very salty water) that contains 30 grams of salt per litre of water is pumped into the tank at a rate of 25 litres per minute. The concentration of salt water after t minutes (in grams per litre) is C(t) = `(30"t")/(200 + "t")`. What happens to the concentration as t → ∞?
Evaluate the following limits:
`lim_(x -> 0) (2^x - 3^x)/x`
Choose the correct alternative:
`lim_(x -> oo) (1/"n"^2 + 2/"n"^2 + 3/"n"^2 + ... + "n"/"n"^2)` is
`lim_(x -> 0) ((2 + x)^5 - 2)/((2 + x)^3 - 2)` = ______.
`lim_(x→-1) (x^3 - 2x - 1)/(x^5 - 2x - 1)` = ______.
`lim_(x→∞)((x + 7)/(x + 2))^(x + 4)` is ______.
