Limit - Mohd Noor
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[email protected]. Mohd Noor Abdul Hamid, Ph.D
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Transcript of Limit - Mohd Noor
To obtain the limit of a function f(x) from the table:
3. Determine the limit for the function as x approach a value: - exist : if the left-side limit = right side limit - do not exist : if the left-side limit ≠ right side limit
2. Build the right side table: - choose a few values (greater than) that x can approach from right - calculate the corresponding output f(x) for the values chosen for x - determine the right-side limit for f(x) based on the table
1. Build the left side table: - choose a few values (less than) that x can approach from left - calculate the corresponding output f(x) for the values chosen for x - determine the left-side limit for f(x) based on the table
Mohd noor abdul hamid : [email protected]
i. Right side limit : approching -1 from right (nilai x > -1); -0.9,-0.99,-0.999….
/f(x)
Mohd noor abdul hamid : [email protected] [email protected]
i. Right-side limit: x approaching -1 from right (x > -1)
When x approach -1 (from right), y is close to 0Thus;
lim f(x) = 0 x-1+
/f(x)
Mohd noor abdul hamid : [email protected] [email protected]
ii) Left-side limit: x approaching -1 from left (x < -1); -1.1,-1.01,-1.001,-1.0001..
/f(x)
Mohd noor abdul hamid : [email protected] [email protected]
ii) Left-side limit: x approaching -1 from left (x < -1)
As x approach -1 (from left), y is getting closer to 0So;
lim f(x) = 0 x-1-
/f(x)
iii) Limit of f(x) as x approach -1:
lim f(x) = lim f(x) = 0 x-1- x-1+
Thus ; lim f(x) = 0 x-1
/f(x)
Mohd noor abdul hamid : [email protected]
i. Right side limit for x0 (x > 0 ; 0.1, 0.01, 0.001,0.001)
As x approach 0 (from right), y is getting closer to 3Thus; lim f(x) = 3 x0+
/f(x)
ii) Left side limit for x0 (x < 0; -0.1,-0.01,-0.001,-0.0001)
When x is approaching 0 (from left), y is getting closer to 3Therefore; lim f(x) = 2 x0-
/f(x)
iii) Limit for f(x) as x approach 0:
lim f(x) ≠ lim f(x) x0- x0+
thus; lim f(x) DOES NOT EXIST x0
c) lim f(x) d) lim f(x) e) lim f(x) x3 x-2 x2
f) lim f(x) g) lim f(x) h) f(0) i) f(2) x-∞ x∞
/f(x)
To obtain limit for a certain function, we only need to substitute the value approaches by x into the function.
EXCEPT : for rational function f(x) = h(x) g(x)
For this case, suppose; lim h(x) = L and lim g(x) = M xa xa
Thus, there are 4 possibilities for the lim f(x) xa
L ≠ 0 and M ≠ 0 thus lim f(x) = L = NON-ZERO xa M
POSSIBILITY 1
To obtain limit for a certain function, we only need to substitute the value approaches by x into the function.
EXCEPT : for rational function f(x) = h(x) g(x)
For this case, suppose; lim h(x) = L and lim g(x) = M xa xa
Thus, there are 4 possibilities for the lim f(x) xa
L = 0 and M ≠ 0 thus lim f(x) = zero = ZERO xa non-zero
POSSIBILITY 2
To obtain limit for a certain function, we only need to substitute the value approaches by x into the function.
EXCEPT : for rational function f(x) = h(x) g(x)
For this case, suppose; lim h(x) = L and lim g(x) = M xa xa
Thus, there are 4 possibilities for the lim f(x) xa
L ≠ 0 and M = 0 thus lim f(x) = non-zero = DO NOT xa zero EXIST
POSSIBILITY 3
To obtain limit for a certain function, we only need to substitute the value approaches by x into the function.
EXCEPT : for rational function f(x) = h(x) g(x)
For this case, suppose; lim h(x) = L and lim g(x) = M xa xa
Thus, there are 4 possibilities for the lim f(x) xa
L = 0 and M = 0 thus lim f(x) = zero = EXIST / NOT?* xa zero * exist : if the function can be simplified * do not exist : if the function is in the simplest form
POSSIBILITY 4
THE CONCEPT of INFINITY (∞)
i) a + ∞ = ∞ ii) ∞ + a = ∞
iii) a - ∞ = -∞ iv) ∞ - a = ∞
v) a.∞ = ∞ vi) a(-∞) = - ∞
vii) a = 0 viii) ∞ = ∞ ∞ a
viii)∞n = ∞ ; for n>0 x) ∞n = 0; for n<0
xi)∞n = 1; for n=0 xii) = ∞
* a is constant
n
LIMIT for x approaching ∞ (X∞)
To obtain limit for a given function, f(x) (EXCEPT for RATIONAL function)as x approaching the value +∞ or -∞, - simply substitute x with the value +∞ or -∞ into the function.
Example:
a) Find lim x3 + 4x2 - 3 x∞
Solution : substitute x = ∞, thus
lim x3 + 4x2 – 3 = ∞3 + 4∞2 - 3x∞
= ∞ + ∞ - 3
= ∞ - 3
= ∞
EXERCISE: Find
a) lim 3 – 4 b) lim -23 x∞ x x∞
c) lim 3x4 + 2x – 1 d) lim 4 – x2 x∞ x∞
LIMIT for x approaching ∞ (X∞)
For RATIONAL function: f(x) = h(x) g(x)
To obtain the limit of a rational function as x approaching +∞ or -∞ ;
i. Eliminates all terms, EXCEPT for the term with highest power in the numerator f(x) and denominator g(x).
ii. Simplify the remaining termsiii. Substitute x with ∞iv. Find the limit for the function.
LIMIT for x approaching ∞ (X∞)
Example 1 : Find lim 2x4 + x2 – 3 x∞ x3 + x + 2
i. Determine which term has the greatest power for numerator (2x4) and denominator (x3)
ii. Eliminates other terms and simplify: lim 2x4 = lim 2x x∞ x3 x∞
iii. substitude x with ∞: lim 2x = 2.∞ x∞
= ∞
For RATIONAL function: f(x) = h(x) g(x)
LIMIT for x approaching ∞ (X∞)
EXERCISE: Find
a)
b)
c)
lim 2x3 + x2 – 3x∞ x3 + x + 2
lim 2x3 + x2 – 3x∞ x4 + x + 2
lim x + 1x∞ 2x + 1
END OF CHAPTER 6
