Math Problem Statement

Find the value

Find the limit: lim(x -> 0) (1 - cos(x)) / x.

Solution

We are given the limit:

limx01cosxx\lim_{{x \to 0}} \frac{1 - \cos{x}}{x}

To evaluate this limit, we can use standard limit techniques. One common approach for this type of problem is applying a known trigonometric limit formula or L'Hopital's Rule, since direct substitution gives an indeterminate form 00\frac{0}{0}.

Method 1: Using Trigonometric Identity

First, we use the trigonometric identity:

1cosx=2sin2(x2)1 - \cos{x} = 2\sin^2\left(\frac{x}{2}\right)

Thus, the given limit becomes:

limx02sin2(x2)x\lim_{{x \to 0}} \frac{2\sin^2\left(\frac{x}{2}\right)}{x}

We now divide both the numerator and denominator by xx:

=limx02sin2(x2)x=2limx0sin2(x2)(x2)12= \lim_{{x \to 0}} \frac{2\sin^2\left(\frac{x}{2}\right)}{x} = 2 \lim_{{x \to 0}} \frac{\sin^2\left(\frac{x}{2}\right)}{\left(\frac{x}{2}\right)} \cdot \frac{1}{2}

Now using the standard limit:

\lim_{{x \to 0}} \frac{\sin{y}}{y} = 1 \quad \text{(where y = \frac{x}{2})},

we get:

2(12)=0.2 \cdot \left( \frac{1}{2} \right) = 0.

Therefore, the value of the limit is 0.

Method 2: Using L'Hopital's Rule

Since the given limit initially results in the indeterminate form 00\frac{0}{0}, we can apply L'Hopital's Rule, which involves taking the derivative of the numerator and the denominator.

  1. Derivative of the numerator 1cosx1 - \cos{x} is sinx\sin{x}.
  2. Derivative of the denominator xx is 11.

Thus, applying L'Hopital's Rule, the limit becomes:

limx0sinx1=sin0=0.\lim_{{x \to 0}} \frac{\sin{x}}{1} = \sin{0} = 0.

So, the value of the limit is 0.

Let me know if you want more details or further clarification!


Here are 5 relative questions that can expand your understanding:

  1. What is L'Hopital's Rule and when should it be applied in limits?
  2. How can we use Taylor series to expand functions like cosx\cos{x} and evaluate limits?
  3. What are some other indeterminate forms besides 00\frac{0}{0}?
  4. Why is the limit limx0sinxx=1\lim_{x \to 0} \frac{\sin{x}}{x} = 1 considered fundamental in calculus?
  5. How does the small-angle approximation sinxx\sin{x} \approx x near zero help in solving limits?

Tip: Memorizing common trigonometric limits and identities can greatly speed up solving problems involving limits like this!

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Math Problem Analysis

Mathematical Concepts

Limits
Trigonometry
L'Hopital's Rule

Formulas

lim(x -> 0) (sin(x) / x) = 1
1 - cos(x) = 2sin^2(x/2)

Theorems

L'Hopital's Rule

Suitable Grade Level

Grade 11-12