De L'Hôpitals rule

[Someonewithoutanswers]

My teacher wrote that: "Given that the limit of f'(x) as x approaches +infinite is 1, because of the De L'Hôpitals rule the limit of f(x)/x as x approaches +infinite is 1 (I understood this part), then the limit of f(x) as x approaches +infinite is +infinite (I don't understand this conclusion, please help me)"

f0,+infinite) -> R and it is possible to derivate it.

 

[pka]

You are asked to find \(\mathop {\lim }\limits_{x \to \infty } \dfrac{{f(x)}}{x}\) knowing that \(\mathop {\lim }\limits_{x \to \infty } f(x) = \infty ~\&~\mathop {\lim }\limits_{x \to \infty } f'(x) = 1\).
But the first tells us that the limit is \(\dfrac{\infty}{\infty}\) so apply l'Hopital's rule to get
\(\mathop {\lim }\limits_{x \to \infty } \dfrac{{f'(x)}}{1} = \dfrac{1}{1} = 1\)

 

[Someonewithoutanswers]

You are asked to find \(\mathop {\lim }\limits_{x \to \infty } \dfrac{{f(x)}}{x}\) knowing that \(\mathop {\lim }\limits_{x \to \infty } f(x) = \infty ~\&~\mathop {\lim }\limits_{x \to \infty } f'(x) = 1\).
But the first tells us that the limit is \(\dfrac{\infty}{\infty}\) so apply l'Hopital's rule to get
\(\mathop {\lim }\limits_{x \to \infty } \dfrac{{f'(x)}}{1} = \dfrac{1}{1} = 1\)

First of all: Thanks for your reply!
Unfortunatly I wasn't asked to find the limit. They just gave me what I wrote in the previous post as something true but I can't understand why "if the limit of f'(x) as x approaches +infinite is 1 then the limit of f(x) as x approaches +infinite is +infinite". They used the rule to justify the statement, but i don't understand how.

 

[Dr.Peterson]

My teacher wrote that: "Given that the limit of f'(x) as x approaches +infinite is 1, because of the De L'Hôpitals rule the limit of f(x)/x as x approaches +infinite is 1 (I understood this part), then the limit of f(x) as x approaches +infinite is +infinite (I don't understand this conclusion, please help me)"

f: (0,+infinite) -> R and it is possible to derivate it.

He's working backward. If L'Hopital's rule applies to f(x)/x, then it must have the form ∞/∞, right? What does that tell you about the limit of f(x)?

To put it differently, if f(x)/x approaches 1 while x approaches ∞, then since the denominator approaches ∞, the numerator must also approach ∞. Isn't that clear? If f(x) approached a finite limit, or had no limit, then f(x)/x would behave differently.

To put it yet another way, the limit of f(x) will be the limit of x times f(x)/x, which is the limit of x ... (Technically, you need to fill in some details about limits existing.)

 

[Someonewithoutanswers]

He's working backward. If L'Hopital's rule applies to f(x)/x, then it must have the form ∞/∞, right? What does that tell you about the limit of f(x)?

To put it differently, if f(x)/x approaches 1 while x approaches ∞, then since the denominator approaches ∞, the numerator must also approach ∞. Isn't that clear? If f(x) approached a finite limit, or had no limit, then f(x)/x would behave differently.

To put it yet another way, the limit of f(x) will be the limit of x times f(x)/x, which is the limit of x ... (Technically, you need to fill in some details about limits existing.)

I didn't understand how to demonstrate "if f(x)/x approaches 1 while x approaches ∞, then since the denominator approaches ∞, the numerator must also approach ∞ " but thinking about it now it is just logic. Thank you so much for your explanation.