L'Hôpital's rule influenced by Stolz–Cesàro theorem
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    {\displaystyle \lim _{x\to c}\left(\liminf _{y\in S_{x}}{\frac {f(y)}{g(y)}}\right)=\liminf _{x\to c}{\frac {f(x)}{g(x)}}}
   and 
  
    
      
        
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    {\displaystyle \lim _{x\to c}\left(\limsup _{y\in S_{x}}{\frac {f(y)}{g(y)}}\right)=\limsup _{x\to c}{\frac {f(x)}{g(x)}}.}
  In case 1, the squeeze theorem establishes that 
  
    
      
        
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    {\displaystyle \lim _{x\to c}{\frac {f(x)}{g(x)}}}
   exists and is equal to L. In the case 2, and the squeeze theorem again asserts that 
  
    
      
        
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    {\displaystyle \liminf _{x\to c}{\frac {f(x)}{g(x)}}=\limsup _{x\to c}{\frac {f(x)}{g(x)}}=L}
  , and so the limit 
  
    
      
        
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    {\displaystyle \lim _{x\to c}{\frac {f(x)}{g(x)}}}
   exists and is equal to L. This is the result that was to be proven.
In case 2 the assumption that f(x) diverges to infinity was not used within the proof. This means that if |g(x)| diverges to infinity as x approaches c and both f and g satisfy the hypotheses of L'Hôpital's rule, then no additional assumption is needed about the limit of f(x): It could even be the case that the limit of f(x) does not exist. In this case, L'Hopital's theorem is actually a consequence of Cesàro–Stolz.In the case when |g(x)| diverges to infinity as x approaches c and f(x) converges to a finite limit at c, then L'Hôpital's rule would be applicable, but not absolutely necessary, since basic limit calculus will show that the limit of f(x)/g(x) as x approaches c must be zero.