Advertisements
Advertisements
Question
Evaluate the following limits:
`lim_(x -> 1) (root(3)(7 + x^3) - sqrt(3 + x^2))/(x - 1)`
Advertisements
Solution
`lim_(x -> 1) (root(3)(7 + x^3) - sqrt(3 + x^2))/(x - 1) = lim_(x -> 1) (root(3)(7 + x^2) - 2 + 2 sqrt(3 + x^2))/(x - 1)`
= `lim_(x -> 1) ((7 + x^3)^(1/3) - 2)/(x - 1) - lim_(x -> 1) ((3 + x^2)^(1/2) - 2)/(x - 1)`
= `lim_(x -> 1) ((7 + x^3)^(1/3) - (8)^(1/3))/(x^3 - 1) xx (x^3 - 1)/(x - 1) - lim_(x -> 1) ((3 + x^2)^(1/2) - (4)^(1/2))/(x^2 - 1) xx (x^2 - 1)/(x - 1)`
= `lim_(x -> 1) ((7 + x^3)^(1/3) - (8)^(1/2))/((7 + x^3) - 8) xx ((x - 1)(x^2 + x + 1))/(x - 1) - lim_(x -> 0) ((3 + x^2)^(1/2) - (4)^(1/2))/((3 + x^2) - 4) xx ((x + 1)(x - 1))/(x - 1)`
= `lim_(x -> 1) ((7 + x^3)^(1/3) - (8)^(1/2))/((7 + x^3) - 8) xx (x^2 + x + 1) - lim_(x _> 1) ((3 + x^2)^(4)^(1/2))/((3 + x^2) - 4) xx (x + 1)`
`lim_(x -> "a") (x^"n" - "a"^"n") = "na"^("n" - 1)`
= `1/3(8)^(1/3 - 1) (1^2 + 1 + 1) - 1/2(4)^(1/2 - 1) (1 + 1)`
= `1/3(8)^(-2/3) (3) - 1/2 xx (4)^(-1/2) xx (2)`
= `(2^3)^(-2/3) - (2^2)^(- 1/2)`
= `2^(-2) - 2^(-1)`
= `1/2^2 - 1/2`
= `1/4 - 1/2`
= `(1 - 2)/4`
= `- 1/4`
APPEARS IN
RELATED QUESTIONS
Evaluate the following limit :
`lim_(x -> 0)[(root(3)(1 + x) - sqrt(1 + x))/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 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 -> 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 -> 1) sin pi 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 -> 0) sec x`
Sketch the graph of a function f that satisfies the given value:
f(– 2) = 0
f(2) = 0
`lim_(x -> 2) f(x)` = 0
`lim_(x -> 2) f(x)` does not exist.
Evaluate the following limits:
`lim_(x ->) (x^"m" - 1)/(x^"n" - 1)`, m and n are integers
Evaluate the following limits:
`lim_(sqrt(x) -> 3) (x^2 - 81)/(sqrt(x) - 3)`
Evaluate the following limits:
`lim_(x -> 0) (sqrt(1 + x) - 1)/x`
Evaluate the following limits:
`lim_(x - 0) (sqrt(1 + x^2) - 1)/x`
Evaluate the following limits:
`lim_(x -> 0) (sqrt(1 - x) - 1)/x^2`
Find the left and right limits of f(x) = tan x at x = `pi/2`
Show that `lim_("n" -> oo) (1^2 + 2^2 + ... + (3"n")^2)/((1 + 2 + ... + 5"n")(2"n" + 3)) = 9/25`
Evaluate the following limits:
`lim_(x-> 0) (1 - cos x)/x^2`
Evaluate the following limits:
`lim_(x - oo){x[log(x + "a") - log(x)]}`
Evaluate the following limits:
`lim_(x -> ) (sinx(1 - cosx))/x^3`
Choose the correct alternative:
`lim_(x - pi/2) (2x - pi)/cos x`
Choose the correct alternative:
`lim_(x -> 0) sqrt(1 - cos 2x)/x`
