Math Problem Statement

What is the limit of tan (2x) when x is approaching pi/4 from the negative

Solution

To find the limit of tan(2x)\tan(2x) as xx approaches π4\frac{\pi}{4} from the negative side, we proceed step by step.

Step 1: Substituting the limit value into 2x2x

As xπ4x \to \frac{\pi}{4}^-, we have: 2x2×π4=π22x \to 2 \times \frac{\pi}{4} = \frac{\pi}{2}

Step 2: Behavior of tan(π2)\tan \left( \frac{\pi}{2} \right)

The tangent function tan(θ)\tan(\theta) becomes undefined at θ=π2\theta = \frac{\pi}{2}. However, we are interested in what happens when θ=2x\theta = 2x approaches π2\frac{\pi}{2} from the negative side.

  • As 2xπ22x \to \frac{\pi}{2}^-, tan(2x)\tan(2x) approaches ++\infty because the tangent function tends to positive infinity when approaching π2\frac{\pi}{2} from the left side.

Conclusion:

limxπ4tan(2x)=+\lim_{x \to \frac{\pi}{4}^-} \tan(2x) = +\infty

Would you like more details or have any further questions?

Related Questions:

  1. What is the limit of tan(2x)\tan(2x) as xπ4+x \to \frac{\pi}{4}^+?
  2. How does the tangent function behave at other critical points, like x=3π2x = \frac{3\pi}{2}?
  3. Can we generalize the behavior of trigonometric limits involving tan(kx)\tan(kx) for any constant kk?
  4. What is the limit of sin(2x)\sin(2x) as xπ4x \to \frac{\pi}{4}^-?
  5. How can we use series expansions to analyze the limit of tan(x)\tan(x) near x=π2x = \frac{\pi}{2}?

Tip:

When dealing with trigonometric limits near critical points, always check whether the function tends toward infinity or negative infinity by considering which side of the point you're approaching from!

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Limits
Trigonometric Functions
Continuity

Formulas

lim_{x \to c^-} \tan(2x)
Behavior of tan(θ) as θ approaches π/2

Theorems

Limit Theorem for Trigonometric Functions
Asymptotic Behavior of Tangent Function

Suitable Grade Level

Grades 11-12