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You have provided the limit expression:

\[
\lim_{{w \to 0^-}} -2 \sec(2w)
\]

Let's break this down and evaluate the limit step by step.

1. The secant function \(\sec(x)\) is defined as \( \frac{1}{\cos(x)} \).
2. We are taking the limit as \(w \to 0^-\), meaning we are approaching 0 from the left (negative) side.

### Step-by-step evaluation:

\[
\lim_{{w \to 0^-}} -2 \sec(2w) = -2 \cdot \lim_{{w \to 0^-}} \sec(2w)
\]

\[
\sec(2w) = \frac{1}{\cos(2w)}
\]

As \(w \to 0^-\), \(2w \to 0\). We know that:

\[
\cos(0) = 1
\]

Thus:

\[
\lim_{{w \to 0^-}} \sec(2w) = \sec(0) = \frac{1}{\cos(0)} = 1
\]

Finally:

\[
\lim_{{w \to 0^-}} -2 \sec(2w) = -2 \cdot 1 = -2
\]

### Conclusion:
The value of the limit is \(-2\).

Would you like further details or clarifications?

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Here are 5 related questions to deepen your understanding:
1. What is the definition of one-sided limits, such as \( w \to 0^- \)?
2. How does the secant function behave as its argument approaches certain critical values?
3. How would the limit change if \( w \) approached 0 from the positive side (\( w \to 0^+ \))?
4. How do you evaluate limits involving trigonometric functions?
5. What are the properties of the cosine function around \( w = 0 \)?

**Tip:** When dealing with trigonometric limits, it's often useful to rewrite the expression using known identities such as \( \sec(x) = \frac{1}{\cos(x)} \).

Calculate the limit as w approaches 0 from the left for the function -2sec(2w).

Explore the step-by-step evaluation of the limit of -2sec(2w) as w approaches 0 from the left. This analysis includes the definition and properties of the secant function and the behavior of limits for trigonometric functions. This problem is suitable for university-level calculus courses.

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