Calculus - Limit : Limit of Polynomials

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Limit of Polynomials

what you'll learn...

Limit of Polynomials

Apply Algebra of Limits
For a function $f (x) = a_{n} x^{n}$ $f(x) = a_nx^n$ $+ a_{n - 1} x^{n - 1}$ $+a_(n-1)x^(n-1)$ $+ \dots$ $+ cdots$ $+ a_{1} x^{1}$ $+ a_1x^1$ $+ a_{0}$ $+ a_0$

$lim_{x \to a} f (x)$ $lim_(x->a) f(x)$

$= a_{n} lim_{x \to a} x^{n}$ $quad quad = a_n lim_(x->a) x^n$ $+ a_{n - 1} lim_{x \to a} x^{n - 1}$ $+a_(n-1) lim_(x->a) x^(n-1)$ $+ \dots$ $+ cdots$ $+ a_{1} lim_{x \to a} x^{1}$ $+ a_1 lim_(x->a) x^1$ $+ a_{0}$ $+ a_0$

limit of a polynomial

Using the algebra of limits, the limit of function $f (x) = 3 x^{3} + 2 x^{2} - 1$ $f(x)=3x^3+2x^2-1$ at $x = - 2$ $x=-2$ is found by applying limit to the three terms of the function.

limit of function $f (x) = 3 x^{3} + 2 x^{2} - 1$ $f(x)=color(deepskyblue)(3x^3)+color(coral)(2x^2)-1$ at $x = - 2$ $x=-2$

By Substitution :
$f (x) ∣_{x = - 2}$ $f(x)|_(x=-2)$
$= 3 {(- 2)}^{3} + 2 {(- 2)}^{2} - 1$ $quad quad = color(deepskyblue)(3(-2)^3)+color(coral)(2(-2)^2)-1$
$= 3 (- 8) + 2 (4) - 1$ $quad quad = color(deepskyblue)(3(-8))+color(coral)(2(4))-1$
$= - 17$ $quad quad = -17$

left-hand-limit :
$lim_{x \to - 2 -} f (x)$ $lim_(x->-2-)f(x)$
$= 3 {(- 2 - δ)}^{3} + 2 {(- 2 - δ)}^{2} - 1$ $quad quad = color(deepskyblue)(3(-2-delta)^3)+color(coral)(2(-2-delta)^2)-1$
substitute $δ = 0$ $delta=0$
$= 3 (- 8) + 2 (4) - 1$ $quad quad = color(deepskyblue)(3(-8))+color(coral)(2(4))-1$
$= - 17$ $quad quad = -17$

right-hand-limit :
$lim_{x \to - 2 +} f (x)$ $lim_(x->-2+)f(x)$
$= 3 {(- 2 + δ)}^{3} + 2 {(- 2 + δ)}^{2} - 1$ $quad quad = color(deepskyblue)(3(-2+delta)^3)+color(coral)(2(-2+delta)^2)-1$
substitute $δ = 0$ $delta=0$
$= 3 (- 8) + 2 (4) - 1$ $quad quad = color(deepskyblue)(3(-8))+color(coral)(2(4))-1$
$= - 17$ $quad quad = -17$

Summary :
limits of function $f (x) = 3 x^{3} + 2 x^{2} - 1$ $f(x)=3x^3+2x^2-1$ at $x = - 2$ $x=-2$

$f (x) ∣_{x = - 2} = - 17$ $f(x)|_(x=-2) = -17$

$lim_{x \to - 2 -} f (x) = - 17$ $lim_(x->-2-)f(x)= -17$

$lim_{x \to - 2 +} f (x) = - 17$ $lim_(x->-2+)f(x)= -17$

All three values are equal. So the function is continuous.

summary

Limit of a polynomial: For a function $f (x) = a_{n} x^{n} + a_{n - 1} x^{n - 1} + \dots + a_{1} x^{1} + a_{0}$ $f(x) = a_nx^n+a_(n-1)x^(n-1)+ cdots + a_1x^1 + a_0$
$lim_{x \to a} f (x)$ $lim_(x->a) f(x)$
$= a_{n} lim_{x \to a} x^{n} + a_{n - 1} lim_{x \to a} x^{n - 1} + \dots$ $quad quad = a_n lim_(x->a) x^n+a_(n-1) lim_(x->a) x^(n-1)+ cdots$
$+ a_{1} lim_{x \to a} x^{1} + a_{0}$ $quad quad quad quad + a_1 lim_(x->a) x^1 + a_0$

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