https://file.aiquickdraw.com/mathbot/file/1726753011872x8zt7ib5.jpg

Let's analyze the image step by step to understand and solve the problems listed below the graph.

The image contains a graph of a function \( h(x) \) and four different limit-related problems. The function has various features, including discontinuities, and the graph is necessary to solve the limits based on the behavior of the function at specific points.

### Problem 1:
\[
\lim_{x \to -2^-} \log_2[h(x)]
\]
This limit asks for the behavior of \( h(x) \) as \( x \) approaches \(-2\) from the left side. From the graph:
- As \( x \to -2^- \), \( h(x) \) approaches \( 0 \).
Since \( \log_2(0) \) is undefined (it tends to \(-\infty\)), this limit tends to:
\[
\lim_{x \to -2^-} \log_2[h(x)] = -\infty.
\]

### Problem 2:
\[
\lim_{x \to 0^+} \frac{21}{e^{h(x)}}
\]
Here we need to evaluate the limit of the fraction as \( x \to 0^+ \). From the graph:
- As \( x \to 0^+ \), \( h(x) \) approaches \( 0 \).
Thus, \( e^{h(x)} \to e^0 = 1 \), and the limit becomes:
\[
\lim_{x \to 0^+} \frac{21}{e^{h(x)}} = \frac{21}{1} = 21.
\]

### Problem 3:
\[
\lim_{x \to -1} h(\tanh^{-1}(-x))
\]
For this limit, we first note the behavior of \( h(x) \) as \( x \to -1 \). From the graph:
- As \( x \to -1 \), \( h(x) \) approaches \( 1 \).
We also know that \( \tanh^{-1}(-x) \to \tanh^{-1}(1) \), which equals \( \infty \). However, looking at the graph, we can deduce that:
\[
\lim_{x \to -1} h(\tanh^{-1}(-x)) = 1.
\]

### Problem 4:
\[
\lim_{x \to +\infty} \left[ \cot^{-1}(h(2x)) + h(3 \coth x) \right]
\]
For this limit:
- As \( x \to +\infty \), \( h(2x) \) and \( h(3 \coth x) \) must be evaluated. From the graph, as \( x \to +\infty \), the function \( h(x) \) tends to zero. Hence:
\[
\lim_{x \to +\infty} \left[ \cot^{-1}(h(2x)) + h(3 \coth x) \right] = \cot^{-1}(0) + 0 = \frac{\pi}{2}.
\]

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### Would you like more detailed explanations for any of these problems? Here are 5 related questions:

1. How can we determine limits for more complex functions from graphs?
2. What happens when limits involve logarithmic or inverse trigonometric functions near undefined values?
3. How do we interpret the behavior of \( h(x) \) at discontinuities?
4. Can limits involving inverse hyperbolic functions be solved analytically?
5. How does the behavior at infinity affect the computation of limits?

**Tip**: Always check both the function's behavior and algebraic simplifications when dealing with limits.

The user uploaded an image containing a graph of a function h(x) and four different limit-related problems, including logarithmic, exponential, inverse trigonometric, and hyperbolic functions.

Analyze the behavior of h(x) and solve limit problems involving logarithmic, exponential, and inverse trigonometric functions using a graph. Learn how limits behave at critical points and discontinuities, with detailed step-by-step solutions. Suitable for advanced high school or college calculus students.

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