#### Attachments

• 491.2 KB Views: 12

#### Subhotosh Khan

##### Super Moderator
Staff member

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:

#### lex

##### Full Member
Consider the real-valued function f defined on $$\displaystyle \mathbb{R}$$ by:

$$\displaystyle f(x) = x + \frac{1}{\sqrt{1+x^2}}$$

1. Calculate: $$\displaystyle \lim \limits_{x \to +\infty} f(x) \text{ and } \lim \limits_{x \to -\infty} f(x)$$
2. Calculate $$\displaystyle f'(x) \text{ for all } x \in \mathbb{R}$$

1. $$\displaystyle \lim \limits_{x \to +\infty} f(x)$$
As $$\displaystyle x \to +\infty \hspace2ex f(x) = x \hspace1ex+$$ a positive term, so $$\displaystyle f(x) \to +\infty$$

$$\displaystyle \lim \limits_{x \to -\infty} f(x)$$
As $$\displaystyle x \to -\infty \hspace2ex f(x) = x\hspace1ex +$$ a positive term less than 1, so $$\displaystyle f(x) \to -\infty$$

2. $$\displaystyle f(x) = x + (1+x^2)^{-\tfrac{1}{2}}$$
Hopefully you can differentiate it in this form?

#### nasi112

##### Full Member
Consider the real-valued function f defined on $$\displaystyle \mathbb{R}$$ by:

$$\displaystyle f(x) = x + \frac{1}{\sqrt{1+x^2}}$$

1. Calculate: $$\displaystyle \lim \limits_{x \to +\infty} f(x) \text{ and } \lim \limits_{x \to -\infty} f(x)$$
2. Calculate $$\displaystyle f'(x) \text{ for all } x \in \mathbb{R}$$

1. $$\displaystyle \lim \limits_{x \to +\infty} f(x)$$
As $$\displaystyle x \to +\infty \hspace2ex f(x) = x \hspace1ex+$$ a positive term, so $$\displaystyle f(x) \to +\infty$$

$$\displaystyle \lim \limits_{x \to -\infty} f(x)$$
As $$\displaystyle x \to -\infty \hspace2ex f(x) = x\hspace1ex +$$ a positive term less than 1, so $$\displaystyle f(x) \to -\infty$$

2. $$\displaystyle f(x) = x + (1+x^2)^{-\tfrac{1}{2}}$$
Hopefully you can differentiate it in this form?
since the problem was in French

Salut lex, bienvenue dans le monde des mathématiques françaises

Hi lex, welcome to the world of French mathematics

By the way, you had a beautiful answer above there, but I think that I have seen this question in the forum before. Maybe yesterday. I mean that there are two identical questions from the same OP. And I think that skeeter has answered the first one.

#### lex

##### Full Member
This is clearly a parallel french universe!
Do I take it you are french?

#### Subhotosh Khan

##### Super Moderator
Staff member
This is clearly a parallel french universe!
Do I take it you are french?
Like English is second language in India, many ex-colonies of France (Vietnam, Cameroon, Syria, Lebanon, etc.) use French.

Last edited:

#### nasi112

##### Full Member
This is clearly a parallel french universe!
Do I take it you are french?
I studied French. I know some French. I am Bahraini. I don't think that you have ever heard of my country.

#### lex

##### Full Member
I studied French. I know some French. I am Bahraini. I don't think that you have ever heard of my country.
Good Heavens. Yes, I have of course heard of the Bahrain Grand Prix!
Like English is second language in India, many ex-colonies of France (Vietnam, Cameroon, Syria, Lebanon, etc.) use French.
True.
I do like French. I studied it myself many years ago in school. I spent some months in Paris about 15 years ago. Hopefully some day travel will be an option again!

#### nasi112

##### Full Member
Good Heavens. Yes, I have of course heard of the Bahrain Grand Prix!

True.
I do like French. I studied it myself many years ago in school. I spent some months in Paris about 15 years ago. Hopefully some day travel will be an option again!
Yup, Grand Prix, formula 1, is a famous event in Bahrain. And let me guess that you are American or maybe English?

#### lex

##### Full Member
Yup, Grand Prix, formula 1, is a famous event in Bahrain. And let me guess that you are American or maybe English?
Neither. Is Éireannach mé. I am Irish.

#### Jomo

##### Elite Member

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at: