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

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 $2$.

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 $3$.

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 $0$.

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 $1$.

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 $\frac{5}{4}$.

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.