Limits

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Introduction and Overview

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Limits

Forms of limits

Limits at infinity

Lim(x->Infinity)f(x)=L

f(x) comes really close to L as it becomes significantly larger, creating a horizontal asymptote at y=L

Lim(x->-Infinity)f(x)=L

f(x) comes really close to L as it becomes significantly more negative, creating a horizontal asymptote at y=L

Lim(x->a)f(x)= -infinity

creates a vertical asymptote at x=a as the values surrounding a, become significantly negative as it approaches a

Lim(x->a)f(x)= infinity

creates a vertical asymptote at x=a as the values surrounding a, become significantly larger as it approaches a

One Sided Limits

Limits approaching form the left

Lim(x->a-)f(x)

Limit of f(x) as x approaches a from the left

Limits approaching from the right

Lim(x->a+)f(x)

Limit of f(x) as x approaches a from the right

Derivative Relationship

Lim(h->0) (f(a+h)-f(a))/h

finding the limit as h approaches 0 is typically the last step within the process when finding the derivative of a function at a certain point

Limits help us to find derivatives of functions

Limit Laws

Fractional Power

Lim(x->12)(3x+1)^1/4

[Lim(x->12)(3x+1)]^1/4

(37)^1/4=2.466

Lim(x->a)(f(x))^n/m

[Lim(x->a)(f(x)]^n/m

Power

Lim(x->4)(4x-3)^2

[Lim(x->4)(4x-3)]^2

(13)^2=169

Lim(x->a)(f(x))^n

[Lim(x->a)(f(x)]^n

Quotient

Lim(x->1)(14x-7)/(8-2x)

Lim(x->1)(14x-7)/Lim(x->1)(8-2x)

=7/6

Lim(x->a)f(x)/g(x)

Lim(x->a)f(x)/Lim(x->a)g(x)

Product

Lim(x->12)(x+9)(x-12)

Lim(x->12)(x+9)*Lim(x->12)(x-12)

21*0=0

Lim(x->a)g(x)f(x)

Lim(x->a)g(x)*Lim(x->a)f(x)

Constant Multiple

Lim(x->4)12(x-9)

12(Lim(x->4)(x-9))

12(-5)=-60

Lim(x->a)cf(x)

c(Lim(x->a)f(x)

Constant

Lim(x->3)89

=89

Lim(x->a)L

=L

Different

Lim(x->3)((x-3)(45-x))

Lim(x->3)(x-3)-Lim(x->3)(45-x)

Sum

Lim(x->3)((x-3)+(45-x))

Lim(x->3)(x-3)+Lim(x->3)(45-x)

0+42=42

Lim(x->a) (f(x)+g(x))

Lim(x->a)f(x)+Lim(x->a)g(x)

Solving

Direct distribution

Only works on certain functions, as long as a is in the domain of f

Transcendental

Lim(x->a)2^x=2^a

Lim(x->3)2^x=2^3

=8

Trig Functions

Lim(x->a)cosx=cosa

Lim(x->0)cosx=cos(0)

=1

Rational

Lim(x->a)1/x-3=1/a-3

Lim(x->8)1/x-3=1/8-3

=1/5

Polynomial

Lim(x->a)x^2-3x+1=a^2-3a+1

Lim(x->4)x^2-3x+1=4^2-3(4)+1

16-12+1=5

For this form you basically would just substitute the number that x is approaching into the function

Solving for surrounding numbers

You can solve for the numbers surrounding that limit within the function, but this only holds true if it comes from the right and left

if Lim(x->3)f(x)

solve for x=2.9999, x=3.001

Defined

intuitive

Lim(x->a) f(x)=L

Limit of f(x) as x approaches a is L

Precise

More has to do with getting as close to the value as possible

Epsilon, along y-axis

room around the y value

Delta, along x-axis

How much room around that x value

Lim(x->a) f(x)=L, if for every number epsilon>0 there is a delta>0

if 0<|x-a|

Continuity

Functions continuous at every number on their domain

root funtions

log functions

inverse trig functions

rational functions

Exponential functions

Trig functions

Polynomials

IVT

if a function is continuous on a closed interval (a,b) and N is a number between f(a) and f(b), as long as those are not equal, then there exist a value c and f(c)=N

DIScontinuous forms

jump discontinuity

limits form the left and right exist but are not the same

lim(x->a)f(x)=DNE

infinite discontinuity

funtion goes to infinity or -infinity at a

lim(x->a)f(x)=infinity (not a real number)

f(a)= may be defined

removable discontinuity

vfff

we can redefine F(x) at a just as a single value

Lim(x->a)f(x)does not = f(a)

f(a)=DNE

lim(x->a)f(x)=L

Requirements

Lim(x->a)f(x)=f(a)

Lim(x->a)f(x) exists

f(a) is defined