Math Problem Statement

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.

Solution

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


26. limt02ttant\lim_{t \to 0} \frac{2t}{\tan t}

For small values of tt, we can use the approximation: tantt as t0.\tan t \approx t \text{ as } t \to 0. Thus, limt02ttant=limt02tt=2.\lim_{t \to 0} \frac{2t}{\tan t} = \lim_{t \to 0} \frac{2t}{t} = 2. The solution is 22.


28. limx06x2(cotx)(csc2x)\lim_{x \to 0} 6x^2 (\cot x) (\csc 2x)

We can use small angle approximations again: cotx=1x,csc2x=12x.\cot x = \frac{1}{x}, \quad \csc 2x = \frac{1}{2x}. Thus, the limit becomes: limx06x2(1x)(12x)=limx06x22x2=3.\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 33.


30. limx0x2x+sinx2x\lim_{x \to 0} \frac{x^2 - x + \sin x}{2x}

We can apply L'Hôpital's Rule since the limit evaluates to 00\frac{0}{0}: f(x)=x2x+sinx,g(x)=2x.f(x) = x^2 - x + \sin x, \quad g(x) = 2x. The derivatives are: f(x)=2x1+cosx,g(x)=2.f'(x) = 2x - 1 + \cos x, \quad g'(x) = 2. Now applying L'Hôpital's Rule: limx02x1+cosx2=1+12=0.\lim_{x \to 0} \frac{2x - 1 + \cos x}{2} = \frac{-1 + 1}{2} = 0. The solution is 00.


32. limh0sin(sinh)sinh\lim_{h \to 0} \frac{\sin (\sin h)}{\sin h}

Using the approximation sinhh\sin h \approx h as h0h \to 0, we also know that sin(sinh)sinhh\sin(\sin h) \approx \sin h \approx h. Thus: limh0sin(sinh)sinh=limh0hh=1.\lim_{h \to 0} \frac{\sin (\sin h)}{\sin h} = \lim_{h \to 0} \frac{h}{h} = 1. The solution is 11.


34. limx0sin5xsin4x\lim_{x \to 0} \frac{\sin 5x}{\sin 4x}

Using the small-angle approximations sin5x5x\sin 5x \approx 5x and sin4x4x\sin 4x \approx 4x, the limit becomes: limx05x4x=54.\lim_{x \to 0} \frac{5x}{4x} = \frac{5}{4}. The solution is 54\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 tantt\tan t \approx t as t0t \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.

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Math Problem Analysis

Mathematical Concepts

Limits
Trigonometric Limits
L'Hopital's Rule
Small Angle Approximations

Formulas

lim (x -> 0) (sin x / x) = 1
L'Hopital's Rule
Small angle approximations: sin x ≈ x, tan x ≈ x

Theorems

L'Hopital's Rule
Trigonometric Limit Theorem

Suitable Grade Level

College Calculus or Advanced High School Calculus