Advertisements
Advertisements
Question
Find the left and right limits of f(x) = `(x^2 - 4)/((x^2 + 4x+ 4)(x + 3))` at x = – 2
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 -> 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 -> 1) (x^2 + x + 1)` = 3
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 |
If the limit of f(x) as x approaches 2 is 4, can you conclude anything about f(2)? Explain reasoning
Evaluate the following limits:
`lim_(x ->) (x^"m" - 1)/(x^"n" - 1)`, m and n are integers
Evaluate the following limits:
`lim_(x -> 0) (sqrt(1 + x) - 1)/x`
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)`
Evaluate the following limits:
`lim_(x -> "a") (sqrt(x - "b") - sqrt("a" - "b"))/(x^2 - "a"^2) ("a" > "b")`
Show that `lim_("n" -> oo) (1 + 2 + 3 + ... + "n")/(3"n"^2 + 7n" + 2) = 1/6`
An important problem in fishery science is to estimate the number of fish presently spawning in streams and use this information to predict the number of mature fish or “recruits” that will return to the rivers during the reproductive period. If S is the number of spawners and R the number of recruits, “Beverton-Holt spawner recruit function” is R(S) = `"S"/((alpha"S" + beta)` where `alpha` and `beta` are positive constants. Show that this function predicts approximately constant recruitment when the number of spawners is sufficiently large
A tank contains 5000 litres of pure water. Brine (very salty water) that contains 30 grams of salt per litre of water is pumped into the tank at a rate of 25 litres per minute. The concentration of salt water after t minutes (in grams per litre) is C(t) = `(30"t")/(200 + "t")`. What happens to the concentration as t → ∞?
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)`
Evaluate the following limits:
`lim_(x -> 0) (sqrt(2) - sqrt(1 + cosx))/(sin^2x)`
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
`lim_(x→-1) (x^3 - 2x - 1)/(x^5 - 2x - 1)` = ______.
