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Find the limit of $\left(\frac{1-\cos\left(x\right)}{x}\right)^2$ as $x$ approaches 0

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Final answer to the problem

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Step-by-step Solution

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Apply the power rule for limits: $\lim_{x\to a}\left(f(x)\right)^n=\left(\lim_{x\to a}f(x)\right)^n$

${\left(\lim_{x\to0}\left(\frac{1-\cos\left(x\right)}{x}\right)\right)}^2$

Learn how to solve limits by direct substitution problems step by step online.

${\left(\lim_{x\to0}\left(\frac{1-\cos\left(x\right)}{x}\right)\right)}^2$

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Learn how to solve limits by direct substitution problems step by step online. Find the limit of ((1-cos(x))/x)^2 as x approaches 0. Apply the power rule for limits: \lim_{x\to a}\left(f(x)\right)^n=\left(\lim_{x\to a}f(x)\right)^n. If we directly evaluate the limit \lim_{x\to 0}\left(\frac{1-\cos\left(x\right)}{x}\right) as x tends to 0, we can see that it gives us an indeterminate form. We can solve this limit by applying L'H么pital's rule, which consists of calculating the derivative of both the numerator and the denominator separately. After deriving both the numerator and denominator, the limit results in.

Final answer to the problem

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Solving a math problem using different methods is important because it enhances understanding, encourages critical thinking, allows for multiple solutions, and develops problem-solving strategies. Read more

Solve using direct substitutionSolve using limit propertiesSolve using L'H么pital's ruleSolve using factorizationSolve using rationalizationSolve without using l'H么pital

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Function Plot

Plotting: $\left(\frac{1-\cos\left(x\right)}{x}\right)^2$

Main Topic: Limits by Direct Substitution

Find limits of functions at a specific point by directly plugging the value into the function.

Used Formulas

See formulas (6)

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