Previous
Next

5. Asímptotes

5.1. Asímptotes verticals

Com podem trobar les asímptotes verticals d'una funció?

Una funció té una asímptota vertical en x = a si   stack bold l bold i bold m with bold x bold rightwards arrow bold a below bold italic f bold left parenthesis bold italic x bold right parenthesis bold equals bold infinity

Majoritàriament, els possibles valors de x on pot passar això són els punts que no són del domini i els punts que sent del domini hi ha algun límit lateral que dóna ∞

Un cop detectats aquests punts cal comprovar que  limit as x rightwards arrow a of f left parenthesis x right parenthesis equals infinity.

Per poder dibuixar la gràfica de la funció al voltant d'aquesta asímptota, cal fer els límits laterals. I en funció del resultat podem saber com van les branques de l'asímptota.

Exemple 1

bold italic f bold left parenthesis bold italic x bold right parenthesis bold equals fraction numerator bold 1 over denominator bold x bold plus bold 1 end fraction 

 Dom(f(x))= straight real numbers minus left curly bracket negative 1 right curly bracket

Mirem si hi ha asímptota en x=-1

limit as x rightwards arrow negative 1 of space fraction numerator 1 over denominator x plus 1 end fraction equals 1 over 0 equals infinity space space rightwards arrow space space space bold italic x bold equals bold minus bold 1 bold space bold space bold italic é bold italic s bold space bold italic A bold. bold italic V

                                                                               

Per determinar els signes de l'infinity, fem els límits laterals:

  open table attributes columnalign right end attributes row cell limit as x rightwards arrow negative 1 to the power of plus of space fraction numerator 1 over denominator x plus 1 end fraction equals 1 over 0 to the power of plus equals plus infinity end cell row cell limit as x rightwards arrow negative 1 to the power of minus of space fraction numerator 1 over denominator x plus 1 end fraction equals 1 over 0 to the power of minus equals negative infinity end cell end table close curly brackets                                                                                                                  

   

                

Exemple 2

bold italic f bold left parenthesis bold italic x bold right parenthesis bold equals fraction numerator bold minus bold x over denominator bold x bold minus bold 2 end fraction 

Dom(f(x))= straight real numbers minus left curly bracket 2 right curly bracket

Mirem si hi ha asímptota en x=2

  limit as x rightwards arrow 2 of fraction numerator negative x over denominator x minus 2 end fraction equals 2 over 0 equals infinity space space rightwards arrow space space space bold italic x bold equals bold 2 bold space bold italic é bold italic s bold space bold italic A bold. bold italic V      

                                                                   

Per determinar els signes de l'∞, fem els límits

laterals: 

open table attributes columnalign right end attributes row cell limit as x rightwards arrow 2 to the power of plus of fraction numerator negative x over denominator x minus 2 end fraction equals fraction numerator negative 2 over denominator 0 to the power of plus end fraction equals negative infinity end cell row cell limit as x rightwards arrow 2 to the power of minus of fraction numerator negative x over denominator x minus 2 end fraction equals fraction numerator negative 2 over denominator 0 to the power of minus end fraction equals plus infinity end cell end table close curly brackets  

   
 

Exemple 3

bold italic f bold left parenthesis bold italic x bold right parenthesis bold equals bold x over open parentheses bold x bold minus bold 2 close parentheses to the power of bold 2 

Dom(f(x))= straight real numbers minus left curly bracket 2 right curly bracket

Mirem si hi ha asímptota en x=2

limit as x rightwards arrow 2 of space x over open parentheses x minus 2 close parentheses squared equals 2 over 0 equals infinity space rightwards arrow space space bold space bold italic x bold equals bold 2 bold space bold italic é bold italic s bold space bold italic A bold. bold italic V             

                          

Per determinar els signes de l'∞, fem els límits laterals:     

open table attributes columnalign right end attributes row cell limit as x rightwards arrow 2 to the power of plus of x over open parentheses x minus 2 close parentheses squared equals 2 over 0 to the power of plus equals plus infinity end cell row cell limit as x rightwards arrow 2 to the power of minus of x over open parentheses x minus 2 close parentheses squared equals 2 over 0 to the power of plus equals plus infinity end cell end table close curly brackets                  

  

Exemple 4

bold italic f bold left parenthesis bold italic x bold right parenthesis bold equals fraction numerator bold minus bold x over denominator open parentheses bold x bold minus bold 2 close parentheses to the power of bold 2 end fraction 

Dom(f(x))= straight real numbers minus left curly bracket 2 right curly bracket

Mirem si hi ha asímptota en x=2

limit as x rightwards arrow 2 of space fraction numerator negative x over denominator open parentheses x minus 2 close parentheses squared end fraction equals fraction numerator negative 2 over denominator 0 end fraction equals infinity space space rightwards arrow space bold italic x bold equals bold 2 bold space bold italic é bold italic s bold space bold italic A bold. bold italic V              

              

 Per determinar els signes de l'∞, fem els límits laterals: 

open table attributes columnalign right end attributes row cell limit as x rightwards arrow 2 to the power of plus of fraction numerator negative x over denominator open parentheses x minus 2 close parentheses squared end fraction equals fraction numerator negative 2 over denominator 0 to the power of plus end fraction equals negative infinity end cell row cell limit as x rightwards arrow 2 to the power of minus of fraction numerator negative x over denominator open parentheses x minus 2 close parentheses squared end fraction equals fraction numerator negative 2 over denominator 0 to the power of plus end fraction equals negative infinity end cell end table close curly brackets                         

  

          

Exemple 5

bold italic f bold left parenthesis bold italic x bold right parenthesis bold equals fraction numerator bold x bold minus bold 2 over denominator bold x to the power of bold 2 bold minus bold 9 end fraction 

Dom(f(x))= straight real numbers minus left curly bracket plus-or-minus 3 right curly bracket

Mirem si hi ha asímptota en x=3   i en x=-3

     x equals 3

   bold lim with bold x bold rightwards arrow bold 3 below space fraction numerator bold x bold minus bold 2 over denominator bold x to the power of bold 2 bold minus bold 9 end fraction equals 1 over 0 equals infinity space space space rightwards arrow space bold italic x bold equals bold 3 bold space bold italic é bold italic s bold space bold italic A bold. bold italic V space    

  Per determinar els signes de l'∞, fem els límits laterals:                                    

open table attributes columnalign right end attributes row cell limit as straight x rightwards arrow 3 to the power of plus of space fraction numerator straight x minus 2 over denominator straight x squared minus 9 end fraction equals 1 over 0 to the power of plus equals plus infinity end cell row cell limit as straight x rightwards arrow 3 to the power of minus of space fraction numerator straight x minus 2 over denominator straight x squared minus 9 end fraction equals 1 over 0 to the power of minus equals negative infinity end cell end table close curly brackets    

     x equals negative 3

 bold lim with bold x bold rightwards arrow bold minus bold 3 below space fraction numerator bold x bold minus bold 2 over denominator bold x to the power of bold 2 bold minus bold 9 end fraction equals 1 over 0 equals infinity space space rightwards arrow space bold italic x bold equals bold minus bold 3 bold space bold italic é bold italic s bold space bold italic A bold. bold italic V space

Per determinar els signes de l'∞, fem els límits laterals:

open table attributes columnalign right end attributes row cell limit as straight x rightwards arrow negative 3 to the power of plus of space fraction numerator straight x minus 2 over denominator straight x squared minus 9 end fraction equals 1 over 0 to the power of minus equals negative infinity end cell row cell limit as straight x rightwards arrow negative 3 to the power of minus of space fraction numerator straight x minus 2 over denominator straight x squared minus 9 end fraction equals 1 over 0 to the power of plus equals plus infinity end cell end table close curly brackets

 
 
 

 

Exemple 6

bold italic f bold left parenthesis bold italic x bold right parenthesis bold equals fraction numerator bold x bold minus bold 1 over denominator bold x to the power of bold 2 bold minus bold 1 end fraction 

Dom(f(x))= straight real numbers minus left curly bracket negative 1 comma space 1 right curly bracket

Mirem si hi ha asímptota en x = 1

limit as x rightwards arrow 1 of fraction numerator x minus 1 over denominator x squared minus 1 end fraction equals 0 over 0 space space space rightwards arrow space space limit as x rightwards arrow 1 of fraction numerator up diagonal strike left parenthesis x minus 1 right parenthesis end strike over denominator up diagonal strike left parenthesis x minus 1 right parenthesis end strike left parenthesis x plus 1 right parenthesis end fraction equals 1 half rightwards double arrow bold italic N bold italic o bold space bold italic h bold italic i bold space bold italic h bold italic a bold space bold italic A bold italic V bold space bold italic e bold italic n bold space bold italic x bold equals bold 1

Mirem si hi ha asímptota en x = -1

limit as x rightwards arrow negative 1 of space fraction numerator x minus 1 over denominator x squared minus 1 end fraction equals fraction numerator negative 2 over denominator 0 end fraction equals infinity space space rightwards arrow space bold italic x bold equals bold minus bold 1 bold space bold italic é bold italic s bold space bold italic A bold italic V       

             

   Per determinar els signes de l'∞, fem els límits laterals:

open table attributes columnalign right end attributes row cell limit as x rightwards arrow negative 1 to the power of plus of fraction numerator x minus 1 over denominator x squared minus 1 end fraction equals fraction numerator negative 2 over denominator 0 to the power of minus end fraction equals plus infinity end cell row cell limit as x rightwards arrow negative 1 to the power of minus of fraction numerator x minus 1 over denominator x squared minus 1 end fraction equals fraction numerator negative 2 over denominator 0 to the power of plus end fraction equals negative infinity end cell end table close curly brackets

                                                                                                         		    
   
Previous
Next