Math Problem Statement

The image contains 14 limit problems involving trigonometric and algebraic functions.

Solution

The image you have uploaded contains a set of 14 limit problems under the title "Latihan Soal." Let me provide solutions to a few of these problems with detailed steps.

Problem 1:

limx0cosxx+1\lim_{x \to 0} \frac{\cos x}{x + 1}

Solution:

  1. As x0x \to 0, cosxcos0=1\cos x \to \cos 0 = 1 and x+11x + 1 \to 1.

Thus, the limit is: limx0cos00+1=11=1\lim_{x \to 0} \frac{\cos 0}{0 + 1} = \frac{1}{1} = 1

Problem 2:

limθπ2θcosθ\lim_{\theta \to \frac{\pi}{2}} \theta \cos \theta

Solution:

  1. As θπ2\theta \to \frac{\pi}{2}, cosθcosπ2=0\cos \theta \to \cos \frac{\pi}{2} = 0.

Thus, the limit is: limθπ2θcosθ=π20=0\lim_{\theta \to \frac{\pi}{2}} \theta \cos \theta = \frac{\pi}{2} \cdot 0 = 0

Problem 3:

limt0cos2t1+sint\lim_{t \to 0} \frac{\cos^2 t}{1 + \sin t}

Solution:

  1. As t0t \to 0, cos2tcos2(0)=1\cos^2 t \to \cos^2(0) = 1 and sint0\sin t \to 0, so 1+sint11 + \sin t \to 1.

Thus, the limit is: limt0cos201+sin0=11=1\lim_{t \to 0} \frac{\cos^2 0}{1 + \sin 0} = \frac{1}{1} = 1

Problem 4:

limx0sinx3xtanx\lim_{x \to 0} \frac{\sin x}{3x \tan x}

Solution:

  1. Use the fact that limx0sinxx=1\lim_{x \to 0} \frac{\sin x}{x} = 1 and limx0tanxx=1\lim_{x \to 0} \frac{\tan x}{x} = 1.

Thus, the limit becomes: limx0sinxx3tanxx=13=13\lim_{x \to 0} \frac{\frac{\sin x}{x}}{3 \frac{\tan x}{x}} = \frac{1}{3} = \frac{1}{3}

Problem 5:

limx0sinx2x\lim_{x \to 0} \frac{\sin x}{2x}

Solution:

  1. Again, use limx0sinxx=1\lim_{x \to 0} \frac{\sin x}{x} = 1.

Thus, the limit becomes: limx0sinx2x=12\lim_{x \to 0} \frac{\sin x}{2x} = \frac{1}{2}


Do you want further explanations or solutions to more of these problems? Here are some relative questions for you:

  1. How do you evaluate limits when the function involves a trigonometric identity?
  2. What are common limit rules used in calculus, especially for trigonometric limits?
  3. How does L'Hôpital's Rule apply to limits with indeterminate forms like 0/00/0?
  4. What is the small-angle approximation for trigonometric functions, and how is it useful in limits?
  5. Can limits always be calculated directly, or do you sometimes need additional techniques (like series expansions)?

Tip: When dealing with trigonometric limits, remembering that limθ0sinθθ=1\lim_{\theta \to 0} \frac{\sin \theta}{\theta} = 1 and limθ0tanθθ=1\lim_{\theta \to 0} \frac{\tan \theta}{\theta} = 1 simplifies many problems.

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Math Problem Analysis

Mathematical Concepts

Limits
Trigonometry
Calculus

Formulas

lim_{x -> 0} (sin x) / x = 1
lim_{x -> 0} (tan x) / x = 1
cos^2(x) + sin^2(x) = 1

Theorems

Limit Theorem
Squeeze Theorem
L'Hopital's Rule

Suitable Grade Level

Grades 11-12, Calculus I