sagnuhh

2020-11-26

Find vert and horz asymptotes, and points of inflection for

$f(x)=\mathrm{ln}(x)/x$

The graph the function

The graph the function

Cullen

Skilled2020-11-27Added 89 answers

Step 1

To use :

Vertical asymptotes :

The graph of

Horizontal asymptotes:

If the degree (the largest exponent ) of the denominator is bigger than the degree of the numerator , the horizontal asymptotes is the

If the degree of the numerator is bigger than the denominator ,there is no horizontal asymptotes.

If the degree of the numerator and denominator are same , the horizontal asymptotes equals the leading coefficient (the coefficient of the largest exponent ) of the numerator divided by the leading coefficient of the denominator.

Inflection point :

An inflection point is a point on the graph at which the the second derivative changes sign.

if

if

Step 2

To find Vertical asymptotes:

Use the definition of vertical asymptotes.

Set the denominator of the given function is zero.

Denominator of the given function is x.

To set the denominator is equal to zero.

To get,

Then, the vertical asymptotes of the given function

Step 3

To find the horizontal asymptotes:

To use the above definition of the horizontal asymptotes.

Let

Here, the degree(the largest exponent) of the denominator is bigger than the degree of the numerator.To get, the horizontal asymptotes is

Then,

The horizontal asymptotes of the function

Step 4

To find the inflection point:

To find the first and second derivative of the given function.

To use the quotient rule.

Then,

Diffrentiate the

To find the inflection point:

To find where the

To get,

Next, to identify the inflection point not in the domain or where f (x) is not continuous.

Also, Domain of the function

To get, f(x) is not dfined at

To get,

Next, to check the sign of

To use the intervals:

Let

a)

Sign: -

Behavior: Concave downward

b)

Sign: 0

Behavior: Inflection point

c)

Sign: -

Behavior: Concave upward

Step 5

Next, plug the inflection point

To get,

To get,

0

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