Profusely basically means to do something in excess or over doing something. It describes something done in a huge amount or largely.
Example 01 : While taking Lucifer to the hospital, he was bleeding profusely.
Ex 02 : She planted a profuse amount of flowers in her garden.
Ex 03 : The old woman thanked us profusely for giving her a lift in the heavy rains.
Ex 04 : After the long match, the football players were sweating profusely.
Ex 05 : Besides feeling weak, Jenifer was shivering profusely despite the medication.
Ex 06 : As Sarah requested profusely, I had to join her for lunch with her father.
Ex 07 : Harry apologized profusely to his annoyed girlfriend, for not returning her call.
Ex 08 : This book is great ! It is profusely illustrated with sufficient images.
Ex 09 : Mary is allergic to flowers, after smelling them she starts sneezing profusely.
Ex 10 : Lisa caught everyone's attention at the party, as she ate in a profuse manner.
Ex 11 : After thinking profusely, I decided to take up the new job offer as it seemed very exciting.