Exigence is what prompted you at the arrival/decision of your topic. This is a certain event that that inspired or influenced an author to express his thoughts in writing.
In the beginning of the semester, the class was introduced to the all the rhetorical elements, and almost everyone thought that it was just another word for purpose. I have to admit that I was in that group too, but as the semester went on and I became more familiarized with all the rhetorical elements and their meanings, I saw how exigence can be useful in my writing strategy. Exigence can be used as a source of inspiration, not for you, but to show your audience how deeply you care about the topic, which ultimately helps in the attraction of your target audience, and a larger audience to recognize your writing skills and persuasiveness. For example, exigences could have been put on display in our argumentative essays, just before our concluding paragraphs.