Understanding limits with Graphs of Numerator and Denominator

Limit of Functions with graph of numerator and denominator

» Slopes at the point x=ax=a decide the limits of f(x)f(x) at x=ax=a.

→ slope of numerator at x=0x=0 is 11

→ slope of denominator at x=0x=0 is 1

→ Both LHL and RHL limits =11=1

example

The limit of a function is computed when the function evaluates to indeterminate value 00 at x=a. Seeing the division in indeterminate value 00, the function can be given as f(x)=fn(x)fd(x) such that fn(x)∣x=a=0 and fd(x)∣x=a=0.

Consider the function f(x)=x2-1x-1. The numerator and denominator evaluate to 0 at x=1. The figure plots the numerator and denominator : numerator is the curve in blue, and denominator is the line in orange.

Given the graphs of numerator and denominator of the function f(x)=x2-1x-1. The vertical purple lines, show 1-δ and 1+δ.

The values of denominator at 1-δ and 1+δ are -δ and +δ respectively.

The values of numerator at 1-δ and 1+δ are -2δ+δ2 and +2δ+δ2 respectively.

Given the graphs of numerator and denominator of the function f(x)=x2-1x-1. The vertical purple lines, show 1-δ and 1+δ.

It is noted that the slope of the line defines the values of numerator at 1-δ and 1+δ.

formal restatement

Given the function f(x)=fn(x)fd(x) such that fn(x)∣x=a=0 and fd(x)∣x=a=0.

f(x)∣x=a+δ

=fn(x)|_(x=a+δ)÷fd(x)|x=a+δ

=[fn(x)∣x=a+δ-fn(a)]

÷[fd(x)∣x=a+δ-fd(a)]

as fn(a)=0 and fd(a)=0.

=fn(x)∣x=a+δ-fn(a)δ

÷fd(x)∣x=a+δ-fd(a)δ

=slopefn(x)∣x=a÷slopefd(x)∣x=a.

summary

**Geometrical representation of Limits: **If f(x)=fn(x)fd(x), where

f(x)∣x=a=00;

fn(x)∣x=a=0 and

fd(x)∣x=a=0, then

• the slopes on the left of x=a define the left-hand-limit and

• the slopes on the right of x=a define the right-hand-limit.

The slopes referred are for the numerator and denominator.

Outline

The outline of material to learn "limits (calculus)" is as follows.

Note : * click here for detailed outline of Limits(Calculus).*

→ __Indeterminate and Undefined__

→ __Indeterminate value in Functions__

→ __Expected Value__

→ __Continuity__

→ __Definition by Limits__

→ __Geometrical Explanation for Limits__

→ __Limit with Numerator and Denominator__

→ __Limits of Ratios - Examples__

→ __L'hospital Rule__

→ __Examining a function__

→ __Algebra of Limits__

→ __Limit of a Polynomial__

→ __Limit of Ratio of Zeros__

→ __Limit of ratio of infinities__

→ __limit of Binomial__

→ __Limit of Non-algebraic Functions__