Start by marinating the salmon fillets. Place the salmon in a shallow dish and pour the lime juice over them. Allow the salmon to marinate for a couple of hours in the refrigerator. This will infuse the fish with a refreshing citrus flavor.
In a small bowl, whisk together the fish broth, the juice of the other lemon, and cornstarch until the cornstarch is fully dissolved. Set aside.
Heat a large skillet over medium-high heat and add the canola oil. Once the oil is hot, carefully place the marinated salmon fillets in the skillet, skin-side down if applicable. Sear the salmon for 3-4 minutes per side, or until it is nicely browned and cooked to your desired level of doneness. Remove the salmon from the skillet and set aside.
In the same skillet, add the fish broth mixture and stir well. Cook the broth over medium heat until it begins to thicken slightly, usually 1-2 minutes.
Stir in the capers, including the vinegar they are preserved in. Continue cooking the sauce for an additional minute, allowing the flavors to meld together.
Return the seared salmon fillets to the skillet, coating them with the sauce. Cook for another minute or two, ensuring the salmon is heated through.