Toughness and the Stack
Yes, the creature would die as a State Based Action.

704.1. State-based actions are game actions that happen automatically whenever certain conditions (listed below) are met. State-based actions don't use the stack.
704.3. Whenever a player would get priority (see rule 116, &quot;Timing and Priority&quot;), the game checks for any of the listed conditions for state-based actions, then performs all applicable state-based actions simultaneously as a single event.

So, the state based effects would get checked after Dismember resolves.
Then, this SBA would happen

704.5f If a creature has toughness 0 or less, it's put into its owner's graveyard. Regeneration can't replace this event.

So, before the Giant Growth would get a chance to resolve, the Llanowar Elf would die.