Tough Limit? Use Standard Limits & Win! | JEE Main Maths ⚡
❓ Question Evaluate the limit: lim x → 0 + tan ( 5 x 1 / 3 ) ln ( 1 + 3 x 2 ) ( tan − 1 ( 3 x ) ) 2 [ e 5 x 4 / 3 − 1 ] 🖼️ Question Image ✍️ Short Solution This is a standard JEE-type limit based on small-angle and small-x approximations . As x → 0 + x \to 0^+ , each function behaves like its first-order term . We’ll convert every function into its leading approximation and simplify. 🔹 Step 1 — Recall standard limits As t → 0 t \to 0 : tan t ∼ t \tan t \sim t ln ( 1 + t ) ∼ t \ln(1+t) \sim t tan − 1 t ∼ t \tan^{-1} t \sim t e t − 1 ∼ t e^t - 1 \sim t These shortcuts are mandatory for fast JEE solving. 🔹 Step 2 — Apply approximations one by one 1️⃣ Tangent term tan ( 5 x 1 / 3 ) ∼ 5 x 1 / 3 \tan(5x^{1/3}) \sim 5x^{1/3} 2️⃣ Logarithmic term ln ( 1 + 3 x 2 ) ∼ 3 x 2 \ln(1 + 3x^2) \sim 3x^2 3️⃣ Inverse tangent term tan − 1 ( 3 x ) ∼ 3 x So, ( tan − 1 ( 3 x ) ) 2 ∼ ( 3 x ) 2 = 9 x 4️⃣ Exponential term e 5 x 4 / 3 − 1 ∼ ...