Understanding Infinite Limits of Trigonometric and Natural Log Functions

Analyzing Tangent Function Limits

• 💡 The limit of tangent x as x approaches pi/2 from the left approaches positive infinity.
• ⚠️ Conversely, the limit of tangent x as x approaches pi/2 from the right approaches negative infinity.
• 🎯 Because the one-sided limits do not match, the two-sided limit for tangent x at pi/2 does not exist.

Evaluating Cosecant Function Limits

• 🎯 The limit of cosecant x as x approaches 0 from the left approaches negative infinity.
• 🔑 Cosecant x is defined as 1/sin x, and a zero in the denominator leads to an infinite limit.

Exploring Natural Logarithm Limits

• 📈 As x approaches 0 from the right, the limit of ln x approaches negative infinity.
• ⚠️ The natural logarithm function, ln x, is only defined for positive values (domain is 0 to infinity).
• 🚫 Therefore, the left-sided limit for ln x at 0 does not exist, and consequently, the general limit as x approaches 0 for ln x does not exist.