In the mixing bowl of a stand mixer, place the softened Cream Cheese, Heavy Cream, and Sugar, and mix on low speed until blended. Scrape down the sides of the mixing bowl, and blend on medium until smooth and creamy. Add the Eggs, and Vanilla, and blend until smooth. Add the Lime Juice, Lime zest, and Lime Food Coloring Gel, and blend until well mixed.
Remove the mixing bowl from the stand, and gently fold in the Raspberries, just until mixed. Place a small dish of water on the lowest rack of the oven, underneath where the Cheesecake will bake.
Pour the Cheesecake batter into the Spring-form pan on top of the Graham Cracker crust, and place the pan in the oven at 350 degrees for 20 minutes, checking often to make sure the top doesn't get too brown. (If the top is browning too fast, loosely place a piece of foil over the top - tenting the pan, so it doesn't touch the Cheesecake).
Turn down the oven to 300 degrees, and bake for 45 minutes, or until the center resembles Jello when gently shaken. If the center is liquid like, it's not done, and needs to bake longer, but if it wiggles like Jello, it's done. (It will set up in the refrigerator).
Remove the Cheesecake to a wire rack to cool, and cool to room temperature.
Cover with Parchment paper and Foil, and refrigerate overnight.