Cobweb, but not Spider mites?
Although majesty palms are somewhat resistant to pests, a few bugs can cause problems for these hardy palm trees: spider mites, palm aphids, scale insects and other pests. Spider Mites. These tiny arachnids are more often a problem with indoor majesty palm trees than with outdoor trees. Spider mites do not like water, so the dry fronds of indoor palm trees provide a perfect habitat. These tiny mites look like small dots on the undersides of leaves. They also make webs on the leaves and near the trunk of the tree, as noted in your picture. Treat spider mites on indoor majesty palms by placing the plant outdoors and spraying it with a hose. If the plant is too big to move, treat the plant indoors by spraying it with soapy water or a commercial miticide. Outdoor majesty palms may be susceptible to spider mites if the tree is in a dry, dusty location. Treat outdoor palms by hosing them down with water or, if they are small enough to reach the palm fronds, spraying them with soapy water or a miticide. Wipe spider mites of the leaves with a damp cloth. With cold weather I put my plant in the shower. After wiping leaves spray with soapy water or use a agricultural oil purchased at garden stores. Majesty palms like moisture and humidity, which also prevents spider mites, keep soil evenly moist, and provide humidity for your palm. You may purchase a humidifier or a humidity tray. Place a tray that will hold water under the pot, once your soil is initially sufficiently moist maintain this by maintaining water in the pan for humidity.