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_(z -> -3) [sqrt("z" + 6)/"z"]`
Evaluate the following limit :
If `lim_(x -> 5) [(x^"k" - 5^"k")/(x - 5)]` = 500, find all possible values of k.
Evaluate the following :
Given that 7x ≤ f(x) ≤ 3x2 – 6 for all x. Determine the value of `lim_(x -> 3) "f"(x)`
In problems 1 – 6, using the table estimate the value of the limit
`lim_(x -> - 3) (sqrt(1 - x) - 2)/(x + 3)`
| x | – 3.1 | – 3.01 | – 3.00 | – 2.999 | – 2.99 | – 2.9 |
| f(x) | – 0.24845 | – 0.24984 | – 0.24998 | – 0.25001 | – 0.25015 | – 0.25158 |
In problems 1 – 6, using the table estimate the value of the limit
`lim_(x -> 0) (cos x - 1)/x`
| x | – 0.1 | – 0.01 | – 0.001 | 0.0001 | 0.01 | 0.1 |
| f(x) | 0.04995 | 0.0049999 | 0.0004999 | – 0.0004999 | – 0.004999 | – 0.04995 |
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)`
Evaluate the following limits:
`lim_(x -> 2) (1/x - 1/2)/(x - 2)`
Evaluate the following limits:
`lim_(x -> 1) (root(3)(7 + x^3) - sqrt(3 + x^2))/(x - 1)`
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) (sin^3(x/2))/x^2`
Evaluate the following limits:
`lim_(alpha -> 0) (sin(alpha^"n"))/(sin alpha)^"m"`
Evaluate the following limits:
`lim_(x - oo){x[log(x + "a") - log(x)]}`
Evaluate the following limits:
`lim_(x -> 0) ("e"^("a"x) - "e"^("b"x))/x`
Choose the correct alternative:
`lim_(x - pi/2) (2x - pi)/cos x`
If `lim_(x->1)(x^5-1)/(x-1)=lim_(x->k)(x^4-k^4)/(x^3-k^3),` then k = ______.
If `lim_(x -> 1) (x + x^2 + x^3|+ .... + x^n - n)/(x - 1)` = 820, (n ∈ N) then the value of n is equal to ______.
If f(x) is differentiable at x = 1 and `lim_(h → 0) 1/h f(1 + h) = 5`, then f' (1) is equal to ______.
