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The image contains a series of limit problems labeled from **a** to **g**. Let's go through each limit one by one.

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**a.** \[
\lim_{x \to 0} \frac{3x \sin 4x}{\cos x - 1}
\]

To solve this, we can use **L'Hopital's Rule** since both the numerator and denominator tend to 0 as \(x \to 0\).

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**b.** \[
\lim_{x \to \infty} \left( \ln(\sqrt{x}) \right)^{\frac{2}{x}}
\]

This can be rewritten as:
\[
\lim_{x \to \infty} \left( \frac{1}{2} \ln(x) \right)^{\frac{2}{x}}
\]
Now, evaluate the limit as \(x \to \infty\).

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**c.** \[
\lim_{x \to 0^+} x^{\tan x}
\]

This requires us to first find the behavior of \(\tan x\) as \(x \to 0^+\) and use properties of limits of exponential functions.

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**d.** \[
\lim_{x \to \frac{\pi}{2}} (1 + \cos x)^{\tan x}
\]

As \(x \to \frac{\pi}{2}\), \(\cos x \to 0\) and \(\tan x \to \infty\). We need to carefully handle the indeterminate form.

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**e.** \[
\lim_{x \to 0} \frac{\sin x + e^x}{x^{2021}}
\]

This is a higher-order limit problem, where we analyze the behavior of both the numerator and the denominator as \(x \to 0\).

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**f.** \[
\lim_{x \to 0} \left[ \frac{\ln(1 + 3x^2)}{x(e^x - 1)} + \left(1 - \frac{1}{x}\right)^{2x} \right]
\]

This problem involves logarithmic limits and binomial expansions for the second part.

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**g.** \[
\lim_{x \to 0^+} x \ln(e^{2x} - 1)
\]

Use logarithmic properties and the fact that \(e^{2x} - 1\) can be approximated by \(2x\) as \(x \to 0^+\).

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Would you like to go over any of these limits in more detail, or would you like a step-by-step solution for a specific part?

Here are 5 related questions to consider:
1. How do you apply L'Hopital's Rule in limit calculations?
2. What is the behavior of \(\ln(x)\) as \(x \to \infty\)?
3. How do exponential and logarithmic functions affect limits?
4. How can you handle limits with powers, such as \(x^{\tan x}\)?
5. What are common techniques for solving limits involving trigonometric functions?

**Tip:** Always check for indeterminate forms like \(\frac{0}{0}\) or \(\infty/\infty\) to decide whether L'Hopital's Rule applies!

Calculate the limits for the following problems (a) to (g).

Solve a series of limit problems including logarithmic, exponential, and trigonometric functions. Learn how to apply L'Hopital's Rule, properties of logarithmic limits, and techniques for solving indeterminate forms.

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