Limits of a function's derivative and how it relates to a function.

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The discussion centers on the relationship between the limit of a function's derivative and the limit of the function itself as x approaches a point from the left. The initial hypothesis is that if the limit of f'(x) approaches infinity, then the limit of f(x) must also approach infinity, which is suggested to be false. Participants explore counterexamples but struggle to find suitable ones, raising questions about the significance of the limit being at a specific point rather than approaching infinity. A hint regarding the inverse function's derivative provides some clarity on the topic. Ultimately, the conversation emphasizes the complexity of the relationship between a function and its derivative limits.
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Homework Statement



Prove or disprove: if lim x->b (from the left) f'(x) = infinity then lim x->b (from the left) f(x) = infinity


Homework Equations


my gut instinct is that this is false.


The Attempt at a Solution


I have thought of many counterexamples but none of them seem to work. Does it matter that the limit is at a point and not as x-> infinity?
 
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Hint: (f-1)'=1/f'(f-1(x))

What happens if f'(y)=0.
 
ah! that helps a lot. thanks for the hint!
 

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