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I see that you've uploaded a set of limit problems. Let me walk you through solving the key ones:

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### 26. \(\lim_{t \to 0} \frac{2t}{\tan t}\)

For small values of \(t\), we can use the approximation:
\[
\tan t \approx t \text{ as } t \to 0.
\]
Thus,
\[
\lim_{t \to 0} \frac{2t}{\tan t} = \lim_{t \to 0} \frac{2t}{t} = 2.
\]
The solution is \(2\).

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### 28. \(\lim_{x \to 0} 6x^2 (\cot x) (\csc 2x)\)

We can use small angle approximations again:
\[
\cot x = \frac{1}{x}, \quad \csc 2x = \frac{1}{2x}.
\]
Thus, the limit becomes:
\[
\lim_{x \to 0} 6x^2 \left( \frac{1}{x} \right) \left( \frac{1}{2x} \right) = \lim_{x \to 0} \frac{6x^2}{2x^2} = 3.
\]
The solution is \(3\).

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### 30. \(\lim_{x \to 0} \frac{x^2 - x + \sin x}{2x}\)

We can apply L'Hôpital's Rule since the limit evaluates to \(\frac{0}{0}\):
\[
f(x) = x^2 - x + \sin x, \quad g(x) = 2x.
\]
The derivatives are:
\[
f'(x) = 2x - 1 + \cos x, \quad g'(x) = 2.
\]
Now applying L'Hôpital's Rule:
\[
\lim_{x \to 0} \frac{2x - 1 + \cos x}{2} = \frac{-1 + 1}{2} = 0.
\]
The solution is \(0\).

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### 32. \(\lim_{h \to 0} \frac{\sin (\sin h)}{\sin h}\)

Using the approximation \(\sin h \approx h\) as \(h \to 0\), we also know that \(\sin(\sin h) \approx \sin h \approx h\). Thus:
\[
\lim_{h \to 0} \frac{\sin (\sin h)}{\sin h} = \lim_{h \to 0} \frac{h}{h} = 1.
\]
The solution is \(1\).

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### 34. \(\lim_{x \to 0} \frac{\sin 5x}{\sin 4x}\)

Using the small-angle approximations \(\sin 5x \approx 5x\) and \(\sin 4x \approx 4x\), the limit becomes:
\[
\lim_{x \to 0} \frac{5x}{4x} = \frac{5}{4}.
\]
The solution is \(\frac{5}{4}\).

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Would you like any further details on any of these solutions, or help with another part? Here are some related questions to deepen your understanding:

1. How does L'Hôpital's Rule apply to indeterminate forms in limit problems?
2. What are the small-angle approximations for trigonometric functions?
3. Can you explain the approximation \(\tan t \approx t\) as \(t \to 0\)?
4. What is the importance of applying limits in calculus, particularly in physics?
5. How does the sine function behave as the angle approaches zero?

**Tip:** When using limits involving trigonometric functions, small-angle approximations can be very useful for simplifying the expressions.

Solve the limit problems shown in the image, such as lim (t -> 0) (2t / tan t), lim (x -> 0) 6x^2(cot x)(csc 2x), and others.

Learn how to solve various trigonometric limit problems, including using L'Hopital's Rule and small angle approximations. This detailed solution covers limits like lim (t -> 0) (2t / tan t) and lim (x -> 0) 6x^2(cot x)(csc 2x). Suitable for college calculus or advanced high school students.

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