Limits and asymptotes : 12th grade math lesson in PDF.

12th grade math lessons

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1 I.Limit of a function at infinity
- 1.1 1. Finite limit in infinity
2 II. Infinite limit in infinity
- 2.1 2. Infinite limit in a real
3 III. Operations on the limits.
4 IV. Limit of a composite function
- 4.1 1. Compound function
- 4.2 2. Theorem of composition of limits
5 V. Limitations and comparison
- 5.1 1. Comparison theorem
- 5.2 2. The so-called “gendarmes” or “sandwich” framing theorem.

Limits (sum, product, quotient) in a 12th grade math lesson with the study of indeterminate forms. In this lesson, we will conduct a study of horizontal, vertical and oblique asymptotes in the 12th grade for the compulsory education.
Knowledge needed for this chapter:
$\star\,$ Determine the possible limit of a geometric sequence.
$\star\,$ Study the limit of a sum, product or quotient
of two suites.
$\star\,$ Use a comparison or framing theorem
to determine a limit of a sequence.
$\star\,$ Establish (by derivation or not) the variations of a function.

I.Limit of a function at infinity

In this section, is the representative curve of the function f in any plane.

1. Finite limit in infinity

Definition:

Let f be a function defined at least on an interval of $\mathbb{R}$ of the type $%5Da\,;\,+\infty%5B$ .
The function f has limit ℓ in $+\infty$ if any open interval containing ℓ contains all
values of f (x) for x large enough. Then we note: $\lim_{x\to\,+\infty}f\,(x)\,=\,l$ .

Example:

Let f be the function defined on $%5D0\,;\,+\infty%5B$ by $f\,(x)\,=\frac{1}{x}+\,1$ . We have $\lim_{x\to\,+\infty}\,(\,\frac{1}{x}+1\,\,)\,=\,1$ .
Indeed, the inverse of x approaches 0 as x increases.
Let be an open interval I such that $1\in\,I$ . Then f (x) will always be in I for x large enough.
Graphically, as narrow as a band parallel to the line of equation y = 1 may be, and which
contains, there is always a value of x beyond which does not leave this band.

Horizontal asymptote.

The line with equation y = ℓ is horizontal asymptote to

at $+\infty$ if $\lim_{x\to\,+\infty}f\,(x)\,=\,l$ .

Remark:

Analogously, we define $\lim_{x\to\,-\infty}f\,(x)\,=\,l$ which characterizes a horizontal asymptote at in $-\infty$ of equation y = ℓ.

Example:

We have seen previously that $\lim_{x\to\,+\infty}\,(\,\frac{1}{x}+1\,\,)\,=\,1$ . We also have $\lim_{x\to\,-\infty}\,(\,\frac{1}{x}+1\,\,)\,=\,1$ .
Therefore, the line with equation y = 1 is horizontal asymptote to the curve in $+\infty$ and in $-\infty$ .

Property (admitted): finite limits of usual functions in ± $\infty$ .

Let n be a non-zero natural number.
$\lim_{x\to\,+\infty}\frac{1}{\sqrt{x}}=\lim_{x\to\,+\infty}\frac{1}{x^n}=0$

and $\lim_{x\to\,-\infty}\frac{1}{x^n}=0$

II. Infinite limit in infinity

Definition:

The function f has limit $+\infty$ in $+\infty$ if any interval of $\mathbb{R}$ of the type $%5Da\,;\,+\infty%5B$ contains
all values of f (x) for x large enough. Then we note: $\lim_{x\to\,+\infty}f\,(x)\,=\,+\infty$

Example:

Let f be the square root function. We have $\lim_{x\to\,+\infty}\,\sqrt{x}\,=\,+\infty$ .
Indeed, $\sqrt{x}$ becomes as large as we want as x increases.
Let be an open interval $I\,=%5Da\,;\,+\infty%5B$ . Then f (x) will always be in I for x large enough.
Graphically, if we consider the upper boundary half-plane a line of equation
y = a, there is always a value of a beyond which does not leave this half-plane.

Property (admitted): infinite limits of usual functions in ± $\infty$ .

Let n be a non-zero natural number.
$\lim_{x\to\,+\infty}\sqrt{x}=\lim_{x\to\,+\infty}x^n=+\infty$

and $\lim_{x\to\,-\infty}x^n=0\,(+\infty\,\,si\,\,n\,pair\,;\,-\infty\,si\,\,n\,impair\,).$

2. Infinite limit in a real

Definition:

Let f be a function defined on an open interval of $\mathbb{R}$ of type $%5Dx_0\,-\varepsilon\,;\,x_0%5B$ or $%5Dx_0\,;\,x_0+\varepsilon%5B$ .
The function f has limit $+\infty$ in

if any interval of $\mathbb{R}$ of type $%5DA\,;\,+\infty%5B$ contains all
the values of f (x) for x close enough to

. Then we note: $\lim_{x\to\,x_0}f\,(x)\,=\,+\infty$

Definition: vertical asymptote.

The line of equation

is vertically asymptotic to

if $\lim_{x\to\,x_0}f\,(x)\,=\,+\infty$

or $\lim_{x\to\,x_0}f\,(x)\,=\,-\infty$

Property (admitted) : finite limits of usual functions in 0.

Let n be a non-zero natural number.
$\lim_{x\to\,0^+}\frac{1}{\sqrt{x}}=\lim_{x\to\,0^+}\frac{1}{x^n}=+\infty$

and $\lim_{x\to\,0^+}\frac{1}{x^n}=0\,(+\infty\,\,si\,\,n\,pair\,;\,-\infty\,si\,\,n\,impair\,).$

III. Operations on the limits.

Property: limit of a sum, product and quotient of two functions.

IV. Limit of a composite function

1. Compound function

Definition:

Let f be a function defined on E and having values in F, and let g be a function defined on F.
The compound of f followed by g is the function $g\,o\,f$ defined on E by $g\,o\,f\,(x)\,=\,g(\,f\,(x))$ .

Remark:

Do not confuse $g\,o\,f$ and $fo\,g$ which are, in general, different.

2. Theorem of composition of limits

Theorem:

Let h be the composite of the function f followed by g and a, b and c three real or ± $\infty$ .
If $\lim_{x\to\,a}f\,(x)\,=\,b$ and $\lim_{x\to\,b}g\,(x)\,=\,c$ , then $\lim_{x\to\,a}h\,(x)\,=\,c$ .

V. Limitations and comparison

1. Comparison theorem

Theorem:

2. The so-called “gendarmes” or “sandwich” framing theorem.

Theorem:

Let there be two real a and ℓ and three functions f , g and h such that, for x > a, we have $f\,(x)\,\leq\,\,g(x)\,\leq\,\,h(x)$

.
If $\lim_{x\to\,+\infty}f\,(x)\,=\lim_{x\to\,+\infty}h\,(x)\,=\,l$

, then $\lim_{x\to\,+\infty}g\,(x)\,=l$ .

Remark:

As for the previous comparison theorem, we have two theorems
analogous when x tends to – $\infty$ and when x tends to a real .

Example:

Let’s determine the limit in – $\infty$ of $f\,(x)\,=\,\frac{x\,cos\,x\,}{x^2\,+\,1}$ .
The limit of cos x in – $\infty$ is indeterminate. Therefore the one of f (x) too.
However, for any strictly negative real x, $-1\,\leq\,\,cos\,x\,\leq\,\,1$ so $x\,\leq\,\,x\,cos\,x\,\leq\,\,-x$ .
And dividing member by member by $x^2\,+\,1\,>\,0$ we have :
$\frac{x}{x^2+1}\leq\,\,\frac{x\,cos\,x}{x^2+1}\leq\,\,\frac{-x}{x^2+1}$ .

For $x\,\in\,R\,^*$ , $\frac{x}{x^2\,+\,1}=\frac{1}{x+\frac{1}{x}}$ .

Now, $\lim_{x\to\,-\infty}x+\frac{1}{x}=-\infty$ . $\lim_{x\to\,-\infty}\frac{x}{x^2\,+\,1}=\lim_{x\to\,-\infty}\frac{-x}{x^2\,+\,1}=0$

So, according to the gendarme theorem, $\lim_{x\to\,-\infty}\frac{x\,cos\,x\,}{x^2\,+\,1}=0$ .

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I.Limit of a function at infinity

1. Finite limit in infinity

II. Infinite limit in infinity

2. Infinite limit in a real

III. Operations on the limits.

IV. Limit of a composite function

1. Compound function

2. Theorem of composition of limits

V. Limitations and comparison

1. Comparison theorem

2. The so-called “gendarmes” or “sandwich” framing theorem.

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