Understanding limits with the graph of the function

Understanding Limits with Graph of the function

» Values of Function at x=ax=a

→ Evaluated at input f(x)∣x=af(x)∣x=a or f(a)f(a)

→ Left-hand-limit limx→a-f(x)limx→a−f(x)

→ Right-hand-limit limx→a+f(x)limx→a+f(x)

In the previous pages, limit is defined in algebraic form.

In this topic, the function is considered as a graph in a 2D coordinate plane and the meaning of limit is explained.

example

The value of f(x)=x2-1x-1f(x)=x2−1x−1 when x=1 is 00.

On substituting x=1, we get f(1)=00.

The plot of f(x)=x2-1x-1 is shown.

At x=1, the graph breaks and the function does not evaluate to a real number.

Left-hand-limit of f(x)=x2-1x-1 is shown.

At x=1-δ, dotted vertical line is shown.

Applying limit is moving the vertical line towards x=1 and making δ≅0. This is shown as limx→1- in the figure.

limx→1-f(x)

=(1-δ)2-1(1-δ)-1

=1-2δ+δ2-11-δ-1

=-δ(2-δ)-δ

=2-δ

=2 (substituting δ=0)

Right-hand-limit of f(x)=x2-1x-1 is shown.

At x=1+δ, dotted vertical line is shown.

Applying limit is moving the vertical line towards x=1 and making δ≅0. This is shown as limx→1+ in the figure.

limx→1+f(x)

=(1+δ)2-1(1+δ)-1

=1+2δ+δ2-11+δ-1

=δ(2+δ)δ

=2+δ

=2 (substituting δ=0)

Both the limits of f(x)=x2-1x-1 is shown.

The right-hand-limit and left-hand-limits converge to 2.

Limit of a function at x=a is understood as the value of function at x=a, left side of that : x=a-δ, and right side of that : x=a+δ

summary

Limits of a function at x=a are illustrated in the figure.

• Evaluated at input f(x)∣x=a or f(a)

• Left-hand-limit limx→a-f(x)

• Right-hand-limit limx→a+f(x)

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__