Advertisements
Advertisements
प्रश्न
Evaluate the following limits:
`lim_(sqrt(x) -> 3) (x^2 - 81)/(sqrt(x) - 3)`
Advertisements
उत्तर
`lim_(sqrt(x) -> 3) (x^2 - 81)/(sqrt(x) - 3)`
Put `sqrt(x) - y`,
When `sqrt(x) -> 3`,
We have y → 3
`lim_(sqrt(x) -> 3) (x^2 - 81)/(sqrt(x) - 3) = lim_(sqrt(x) -> 3) (((sqrt(x)^2))^2 - 3^4)/(sqrt(x) - 3)`
= `lim_(sqrt(x) -> 3) ((sqrt(x))^4 - 3^4)/(sqrt(x) - 3)`
= `lim_(y -> 3) (y^4 - 3^4)/(y - 3)`
`lim_(x -> "a") (x^"n" - "a"^"n")/(x - "a") = "na"^("n" - 1)`
= `4(3)^(4 -1)`
= 4 × 33
`lim_(sqrt(x) -> 3) (x^2 - 81)/(sqrt(x) - 3)` = 4 × 27
= 108
APPEARS IN
संबंधित प्रश्न
Evaluate the following limit:
`lim_(x -> 2)[(x^(-3) - 2^(-3))/(x - 2)]`
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)(2x + 3)` = 7
Evaluate the following :
`lim_(x -> 0)[x/(|x| + x^2)]`
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 -> 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)`
Verify the existence of `lim_(x -> 1) f(x)`, where `f(x) = {{:((|x - 1|)/(x - 1)",", "for" x ≠ 1),(0",", "for" x = 1):}`
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 -> 3) (x^2 - 9)/(x^2(x^2 - 6x + 9))`
Show that `lim_("n" -> oo) (1 + 2 + 3 + ... + "n")/(3"n"^2 + 7n" + 2) = 1/6`
Evaluate the following limits:
`lim_(x -> oo)(1 + 1/x)^(7x)`
Evaluate the following limits:
`lim_(x -> oo) ((2x^2 + 3)/(2x^2 + 5))^(8x^2 + 3)`
Evaluate the following limits:
`lim_(x -> 0) (sin^3(x/2))/x^2`
Evaluate the following limits:
`lim_(x -> 0) (sinalphax)/(sinbetax)`
Evaluate the following limits:
`lim_(x -> 0) (sqrt(2) - sqrt(1 + cosx))/(sin^2x)`
Choose the correct alternative:
`lim_(x -> oo) sinx/x`
If `lim_(x->1)(x^5-1)/(x-1)=lim_(x->k)(x^4-k^4)/(x^3-k^3),` then k = ______.
`lim_(x→∞)((x + 7)/(x + 2))^(x + 4)` is ______.
If f(x) is differentiable at x = 1 and `lim_(h → 0) 1/h f(1 + h) = 5`, then f' (1) is equal to ______.
