Limits and Computational Approach
Limits and Computational Techniques- Algebraic techniques for finding limits
- How to use limits of basic function to compute limits of complicated function.
Some basic limit for
f(x) = k The constant function
g(x) = x Straight line through origin
f(x) = k
Limit | Example |
lim_{x→ a} k = k | lim_{x→ 2} 3 = 3, lim_{x→-2} 3 = 3 |
lim_{x→ +∞} k = k | lim_{x→ +∞} 3 = 3, lim_{x→ +∞} 0 = 0 |
lim_{x→ -∞} k = k | lim_{x→ -∞} 3 = 3, lim_{x→-∞} 0 = 0 |
Now for g(x) = x
Limit |
lim_{x→ a} x = a |
lim_{x→ +∞} x = +∞ |
lim_{x→ -∞} x = -∞ |
lim_{x→ a} k = k
lim_{x→ +∞} k = k lim_{x→ -∞} k = k
(a) y = x
lim_{x→ -∞} x = -∞
(b) y = x
lim_{x→ +∞} x = +∞
(c)
Theorem
Let lim stand for one of the limits
lim_{x→ a}, lim_{x→ a-}, lim_{x→ a+}, lim_{x→ +∞} or lim_{x→ -∞}
If L_{1} = lim f(x) and L_{2} = lim g(x) both exist, then
a) lim [f(x) + g(x)] = lim f(x) + lim g(x) = L_{1} + L_{2}
b) lim [f(x) - g(x)] = lim f(x) - lim g(x) = L_{1} - L_{2}
c) lim [f(x).g(x)] = lim f(x).lim g(x) = L_{1}.L_{2}
d) lim [f(x)/g(x)] = [lim f(x)/lim g(x)] = L_{1}/L_{2}
e) lim ^{n}√f(x) = ^{n}√lim f(x) = ^{n}√L_{1}
L_{1} ≥ 0 if n is even
Remark
Although the results (a) and (c) are stated for two functions f and g, these results hold as well for and finite number of functions; that is, if the limits lim f_{1}(x), Lim f_{2}(x),……….lim f_{n}(x) all exists, then
lim [f_{1}(x) + f_{2} + ... + f_{n}(x)] = lim f_{1}(x) + lim f_{2}(x) + ... + lim f_{n}(x)
and
lim [f_{1}(x).f_{2}(x). ... .f_{n}(x)] = lim f_{1}(x).lim f_{2}(x). ... .lim f_{n}(x)
If f_{1},f_{2},………..,f_{n} are same functions
lim [f(x)]^{n} = [lim f(x)]^{n}
Thus we can write
A) lim _{x → a} x^{n} = [lim _{x → a} x]^{n} = a^{n}
Another useful result
B) lim [k.g(x)] = lim (k).lim g(x) = k.lim g(x)
Where k is constant
Polynomial
A polynomial is an expression of the form
f(x) = b_{n}x^{n} + b_{n - 1}x^{n - 1} + ... + b_{1}x + b_{0}
Where b_{n} , b_{n - 1},,…. , b_{1} , b_{0} are all constants.
Example
lim_{x → 5} (x)^{2} - 4x + 3) = lim_{x → 5} x^{2} - 4lim_{x → 5} x + lim_{x → 5} 3 = (5)^{2} - 4(5) + 3 = 8
Theorem 2.5.2
For any polynomial
p(x) = c_{0} + c_{1}x + ... + c_{n}x^{n}
and any real number a
lim _{x → a} p(x) = c_{0} + c_{1}a + ... + c_{n}a^{n} = p(a)
Proof
lim_{x → a} p(x) = lim_{x → a} (c_{0} + c_{1}x + ... + c_{n}x^{n}) = lim_{x → a}c_{0} + lim_{x → a}c_{1}x + ... + lim_{x → a}c_{n}x^{n} = lim_{x → a}c_{0} + c_{1}lim_{x → a} x + ... + c_{n}lim_{x → a} x^{n} = c_{0} + c_{1}a + ... + c_{n}a^{n} = p(a)
Limit involving 1/x
The following limits are suggested by the graph of 1/x.
Table of numerical values
Values | Conslusion | |
x 1/x |
1 .. 0.1 .. 0.001 .. 1 .. 100 .. 1000 .. |
x → 0+ 1/x → +∞ |
x 1/x |
-1 .. -0.01 .. -0.001 -1 .. -100 .. -1000 .. |
x → 0- 1/x → -∞ |
x 1/x |
1 .. 100 .. 1000 .. 1 .. 0.01 .. 0.001 .. |
x → +∞ 1/x decreases towards 0 |
x 1/x |
-1 .. -100 .. -1000 .. -1 .. -0.01 .. -0.001 .. |
x → -∞ 1/x increases towards 0 |
For every real number a the graph of the function g(x) = 1/(x - a) is a translation of f(x) = 1/x by a units to the right.
lim_{x → a-} g(x) = -∞ lim_{x → a+} g(x) = +∞
lim_{x → -∞} g(x) = 0 lim_{x → +∞} g(x) = 0
Limits of polynomiaks as x goes to +∞ and -∞
Lim_{x → +∞} x = +∞ Lim_{x → +∞} x^{2} = +∞
lim_{x → +∞} x^{3} = +∞
Example
lim_{x → +∞} 2x^{5} = 2lim_{x → +∞} x^{5} = +∞
lim_{x → +∞} -7x^{6} = -7lim_{x → +∞} x^{6} = -∞
For integer value of n
lim_{x → +∞} 1/x^{n} = (lim_{x → +∞} 1/x)^{n} = 0
lim_{x → -∞} 1/x^{n} = (lim_{x → -∞} 1/x)^{n} = 0
y = f(x) = 1/x^{n} (n is a positive odd integer)
A polynomial behaves like its term of highest degree as x→ +∞ or x→ -∞ more precisely, if c_{n} = 0 , then
lim_{x → +∞} (c_{0} + c_{1}x + ... + c_{n}x^{n}) = lim_{x → +∞} c_{n}x^{n}
lim_{x → -∞} (c_{0} + c_{1}x + ... + c_{n}x^{n}) = lim_{x → -∞} c_{n}x^{n}
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c_{0} + c_{1}x + ... + c_{n}x^{n} = x^{n}(c_{0}/x^{n} + c_{1}/x^{n - 1} + .. + c_{n}) = c_{n}x^{n}
Example
lim_{x → 2} (5x^{3} + 4)/(x - 3) = [lim_{x → 2} (5x^{3} + 4)]/[lim_{x → 2} (x - 3)] = [5(2)^{3} + 4]/[2 - 3] = -44
The graph has not value at x = 2
Example
lim_{x → 4+} (2 - x)/(x - 4)(x + 2)
lim_{x → 4+} (2 - x)/(x - 4)(x + 2) = -∞
Quick method for finding limit of rational functions