How did the Declaration of Independence change the nature of the American Revolution?
It did not change the nature of the revolution. It was an announcement to the world that we considered ourselves a sovereign nation.
Until July 4, 1776 we were, as far as the world was concerned, a part of the United Kingdom. From that day forward we were autonomous and at war with a foreign country. Until that day we had been in a civil war with our mother country. But the final two statements were entirely reliant upon the war's outcome. If we had lost, we would still have been a colony of Great Britain and would have had to endure punishment for having taken arms against the king.
The early portions of the Declaration go to great lengths to delineate the purported crimes of the king upon America. It was our justification for both the war and our separation. It additionally was meant for the French as Benjamin Franklin had been working hard to gain the support. He had failed that far.
France was not on good terms with England anyway and with America calling itself a nation unto itself, it gave France what it needed to send support.
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The Declaration of Independence changed the nature of the American Revolution by formally declaring the colonies' independence from British rule. It shifted the focus of the revolution from merely fighting against British policies to striving for full independence and self-governance. The Declaration also laid out the philosophical and ideological principles underlying the revolution, such as the belief in natural rights and the idea that governments derive their powers from the consent of the governed. This document not only united the colonies in their quest for independence but also inspired movements for liberty and self-determination around the world.
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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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