Advertisements
Advertisements
Question
Find the left and right limits of f(x) = `(x^2 - 4)/((x^2 + 4x+ 4)(x + 3))` at x = – 2
Advertisements
Solution
f(x) = `(x^2 - 4)/((x^2 + 4x+ 4)(x + 3))` at x = – 2
f(x) = `((x + 2)(x - 2))/((x + 2)^2 (x + 3))`
f(x) = `(x - 2)/((x +2)(x +3))`
o find the let imit of f(x) at x = – 2
Put x = – 2 – h
Where h > 0
When x → – 2
We have h → 0
`lim_(x -> - 2^-) f(x) = lim_("h" -> 0) ((-2 - "h")- 2)/((-2 "h" + 2)(- 2 - "h" + 3)`
= `lim_("h" -> 0) (-4 - "h")/((- "h")(1 - "h"))`
=`lim_("h" -> 0) 1/"h" ((4 + "h")/(1 - "h"))`
= `1/0 ((4 + 0)/(1 - 0))`
= `oo`
`lim_(x -> - 2^-) f(x) = oo`
o find the right limit of f(x) at x = – 2
Put x = – 2 + h
Where h > 0
When x → – 2
We have h → 0
`lim_(x -> - 2) f(x) = lim_("h" -> 0) ((-2 + "h") - 2)/((-2 + "h" + 2)(-2 + "h" + 3))`
= `lim_("h" -> 0) (-4 + "h")/("h"(1 + "h"))`
= `lim_("h" -> 0) 1/"h"(("h" - 4)/(1 +"h"))`
= `1/0 ((0 - 4)/(1 + 0))`
= `- oo`
`lim_(x -> - 2^-) f(x) = - oo`
APPEARS IN
RELATED QUESTIONS
Evaluate the following limit :
`lim_(x -> 7)[((root(3)(x) - root(3)(7))(root(3)(x) + root(3)(7)))/(x - 7)]`
In problems 1 – 6, using the table estimate the value of the limit
`lim_(x -> 0) (sqrt(x + 3) - sqrt(3))/x`
| x | – 0.1 | – 0.01 | – 0.001 | 0.001 | 0.01 | 0.1 |
| f(x) | 0.2911 | 0.2891 | 0.2886 | 0.2886 | 0.2885 | 0.28631 |
In problems 1 – 6, using the table estimate the value of the limit
`lim_(x -> 0) sin x/x`
| x | – 0.1 | – 0.01 | – 0.001 | 0.001 | 0.01 | 0.1 |
| f(x) | 0.99833 | 0.99998 | 0.99999 | 0.99999 | 0.99998 | 0.99833 |
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)`
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`
Sketch the graph of f, then identify the values of x0 for which `lim_(x -> x_0)` f(x) exists.
f(x) = `{{:(x^2",", x ≤ 2),(8 - 2x",", 2 < x < 4),(4",", x ≥ 4):}`
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 f(2) = 4, can you conclude anything about the limit of f(x) as x approaches 2?
Evaluate the following limits:
`lim_(x -> oo) 3/(x - 2) - (2x + 11)/(x^2 + x - 6)`
Evaluate the following limits:
`lim_(x -> oo) (x^4 - 5x)/(x^2 - 3x + 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 + 3/x)^(x + 2)`
Evaluate the following limits:
`lim_(x -> 0) (sinalphax)/(sinbetax)`
Evaluate the following limits:
`lim_(x -> pi) (1 + sinx)^(2"cosec"x)`
Choose the correct alternative:
`lim_(x -> 0) ("a"^x - "b"^x)/x` =
Choose the correct alternative:
`lim_(x -> 0) (x"e"^x - sin x)/x` is
Choose the correct alternative:
`lim_(x -> 0) ("e"^tanx - "e"^x)/(tan x - x)` =
The value of `lim_(x rightarrow 0) (sqrt((1 + x^2)) - sqrt(1 - x^2))/x^2` 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 ______.
