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Function and limit mind map
This is a mind map about functions and limits, including the limits of functions, limit operation rules, continuity of functions, etc. Hope this helps!
Edited at 2023-11-05 19:41:58
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Functions and limits
basic elementary functions
Constant function: y=c (c is a constant)
Power function: y=x^a
Exponential function: y=a^x (a>0 and a≠1)
Logarithmic function: y=logax (a>0 and a≠1)
Trigonometric functions
Sine function: y=sinx
Cosine function: y=cosx
Tangent function: y=tanx
Cotangent function: y=cotx
Secant function: y=secx
Cosecant function: y=cscx
limit of function
Definition: Suppose the function f(x) is defined in a certain decentered domain U of point x0. If when x→x0, the function value f(x) can approach infinitely to a certain constant A, then it is said that when x→ When x0, the limit of function f(x) is A, which is recorded as lim[x→x0]f(x)=A.
Theorem: The necessary and sufficient condition for the existence of the limit of function f(x) when x→x0 is that the left limit and the right limit exist simultaneously and are equivalent, that is, lim[x→x0-]f(x)=lim[x→x0 ]f( x)
Infinitely small and infinitely large
infinitesimal amount
Definition: A variable whose limit is zero is called an infinitesimal quantity, or infinitesimal for short.
lim[x→x0]f(x)=A→0
nature
The algebraic sum of a finite number of infinitesimal quantities is still an infinitesimal quantity
The product of a finite number of infinitesimal quantities is still an infinitesimal quantity
The product of a constant and an infinitesimal quantity is an infinitesimal quantity
The product of a bounded variable and an infinitesimal quantity is an infinitesimal quantity
Compare
Higher order infinitesimal/lower order infinitesimal
If β/α=0, then β is said to be the higher-order infinitesimal of α, denoted as β=0(α).
(Example: when x→0, x³ is the higher-order infinitesimal of x².)
infinitesimal of the same order
If limβ/α=C (C≠0), then β and α are said to be infinitesimals of the same order.
(Example: y=2x and y=x are infinitesimals of the same order)
Equivalent to infinitesimal
Especially when C=1, β and α are said to be equivalent to infinitesimals. Denoted as α~β.
Substitution formula (when x → 0)
sinx~x
arcsinx~x
tanx~x
arctanx~x
e^x-1~x
ln(1 x)~x
1-cosx~1/2x²
(1 αx)^β~αβx
infinite amount
Definition: During a certain change process of the independent variable .
(Example: when x→0, 1/x, 1/x², 1/sinx, 1/tanx are all infinite quantities)
(Example: when x→∞, x², e^x, ln(x 1) are all infinite quantities.)
relationship with infinitesimal quantities
In the same process of change, the reciprocal of infinity is infinitesimal
The reciprocal of infinitesimals that is always not equal to zero is infinity
extreme algorithm
Theorem: Suppose limf(x)=A, limg(x)=B, then we have
The four arithmetic rules still hold
lim[f(x)±g(x)]=limf(x)±limg(x)=A B
lim[f(x)g(x)]=limf(x)·limg(x)=A×B
limf(x)/g(x)=limf(x)/limg(x)=A/B（B≠0）
Corollary: Let limf(x)=A
If C is a constant, then lim[Cf(x)]=Climf(x)=CA
If n is a positive integer, then lim[f(x)]ⁿ=Aⁿ
extreme operation
0/0 formula solution method
factoring
rationalize
Eliminate terms with a denominator of 0 to facilitate calculations
∞/∞ formula solution method
Divide the numerator and denominator simultaneously by the highest power of x
Two important limits
lim[x→0]sinx/x=1
lim[x→0](1 x)^1/x=e or lim[x→∞](1 1/x)^x=e
continuity of function
The function f(x) is continuous at point x0
Increment: △x and △y
Definition: lim[x→x0]f(x)=f(x0)
Determine successive steps
f(x) is defined at point x0
lim[x→0]f(x) exists
lim[x→x0]f(x)=f(x0)
Continuity of a function on an interval
left continuous
right continuous
The function is discontinuous or discontinuous
Definition: If the function f(x) satisfies one of the following conditions at point x0, then the point x0 is called the discontinuity point or discontinuity point of the function f(x).
f(x) is not defined at point x0
lim[x→x0]f(x) does not exist
lim[x→x0]f(x)≠f(x0)
Classification of discontinuities
The first type of discontinuity point (both left and right limits exist)
Discontinuity points can be removed: if lim[x→x0-]f(x) and lim[x→x0 ]f(x) both exist, and lim[x→x0-]f(x)=lim[x→x0 ] f(x), but f(x) is not defined at x0. Then the point x0 is called the discontinuous point of f(x).
Jump discontinuity point: If both lim[x→x0-]f(x) and lim[x→x0 ]f(x) exist, but lim[x→x0-]f(x)≠lim[x→x0 ]f (x), then point x0 is called the jump discontinuity point of function f(x).
The second type of discontinuity point (at least one of the left or right limits does not exist)
Infinite discontinuity point: If limf(x)=∞, or lim[x→x0-]f(x)=∞, or lim[x→x0]f(x)=∞, then the point x0 is called the function f(x ) of infinite discontinuities.
Oscillation discontinuity point: If when x→x0, the function value f(x) changes between two different numbers infinitely, then the point x0 is called the oscillation discontinuity point of the function f(x).
zero point theorem
Definition: When y=0, find the value of x
Judgment steps: f(a)·f(b)<0