Advertisements
Advertisements
Question
Evaluate the following limits:
`lim_(x -> 0) (sqrt(x^2 + "a"^2) - "a")/(sqrt(x^2 + "b"^2) - "b")`
Advertisements
Solution
We know `lim_(x -> "a") (x^"n" - "a"^"a")/(x -"a") = "na"^("n" - 1)`
`lim_(x -> 0)((sqrt((x^2 + "a"^2)) - "a")/(sqrt((x^2 + "b"^2)) - "b")) = lim_(x -> 0)(((x^2 + "a"^2)^(1/2) - ("a"^2)^(1/2))/((x^2 + "b"^2)^(1/2) - ("b"^2)^(1/2)))`
= `lim_(x -> 0) ((x^2 + "a"^2)^(1/2) - ("a"^2)^(1/2))/x^2 xx x^2/((x^2 + "b"^2)^(1/2) - ("b"^2)^(1/2))`
= `lim_(x -> 0) ((x^2 + "a"^2)^(1/2) - ("a"^2)^(1/2))/((x^2 + "a"^2) - "a"^2) xx ((x^2 + "b"^2) - "b"^2)/((x^2 + "b"^2)^(1/2) - ("b"^2)^(1/2))`
= `lim_(x -> 0) ((x^2 + "a"^2)^(1/2) - ("a"^2)^(1/2))/((x^2 + "a"^2) - "a"^2) xx lim_(x -> 0) 1/(((x^2 + "b"^2)^(1/2) - ("b"^2)^(1/2))/((x^2 + "b"^2) - "b"^2)`
Put x2 + a2 = y
Put x2 + b2 = z
When x = 0
⇒ y = a2
When x = 0
⇒ z = b2
= `lim_(y -> "a"^2)((y^(1/2) - ("a"^2)^(1/))/(y - "a"^2)) xx 1/(lim_(x -> "b"^2)(("z"^(1/2) - ("b"^2)^(1/2))/("z" - "b"^2))`
= `1/2 ("a"^2)^(1/2 - 1) xx 1/(1/2 ("b"^2)^(1/2 - 1))`
= `(("a"^2)^(- 1/2))/(("b"^2)^(- 1/2))`
= `("a"^(- 1))/("b"^(- 1))`
= `"b"/"a"`
`lim_(x -> 0)((sqrt((x^2 + "a"^2)) - "a")/(sqrt((x^2 + "b"^2)) - "b")) = "b"/"a"`
APPEARS IN
RELATED QUESTIONS
Evaluate the following limit:
`lim_(y -> -3) [(y^5 + 243)/(y^3 + 27)]`
Evaluate the following limit:
`lim_(z -> -5)[((1/z + 1/5))/(z + 5)]`
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 :
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)`
Evaluate the following :
`lim_(x -> 1) [(x + 3x^2 + 5x^3 + ... + (2"n" - 1)x^"n" - "n"^2)/(x - 1)]`
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 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)`
Evaluate the following limits:
`lim_(x -> 3) (x^2 - 9)/(x^2(x^2 - 6x + 9))`
Evaluate the following limits:
`lim_(x ->oo) (x^3/(2x^2 - 1) - x^2/(2x + 1))`
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) (1 - cos x)/x^2`
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^+)(int_0^(x^2)(sinsqrt("t"))"dt")/x^3` is equal to ______.
