**Core Concept**
Atopic dermatitis, also known as eczema, is a chronic inflammatory skin condition characterized by dry, itchy, and scaly skin lesions. It is often associated with a genetic predisposition and can be exacerbated by environmental factors. The condition is commonly seen in individuals with a family history of allergies, such as asthma or allergies to food, dust, or pollen.

**Why the Correct Answer is Right**
The presentation of erythematous lesions on the cheek and extensor aspect of upper and lower limbs in an infant, combined with a family history of asthma, is highly suggestive of atopic dermatitis. This condition often starts in infancy and is associated with skin barrier dysfunction, leading to increased permeability and inflammation. The involvement of the face, particularly the cheeks, is a classic feature of atopic dermatitis in infants. The presence of asthma in the family further supports a diagnosis of atopic dermatitis, as it is often seen in individuals with a strong atopic predisposition.

**Why Each Wrong Option is Incorrect**
**Option A:** Air borne contact dermatitis typically presents with a more localized reaction to specific allergens, such as pollen or dust mites, and is not typically associated with a family history of asthma.

**Option C:** Seborrheic dermatitis is characterized by a distinct greasy appearance and is often seen in areas rich in sebaceous glands, such as the scalp and face. It is not typically associated with a family history of asthma.

**Option D:** Infectious eczematoid dermatitis, also known as impetigo, is a bacterial infection that presents with characteristic golden crusts and is not typically associated with a family history of asthma.

**Clinical Pearl / High-Yield Fact**
Atopic dermatitis is a chronic condition that often requires a multi-faceted approach to management, including topical corticosteroids, moisturizers, and avoidance of triggers. Early recognition and treatment can help prevent long-term complications and improve quality of life for affected individuals.

**✓ Correct Answer: B. Atopic dermatitis**