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Limit of sin(x)/x as x approaches 0
Published on October 19, 2024
Calculus
The limit of sin(x)/x as x approaches 0 is 1, determined using L'Hôpital's rule.
Question
Calculate the limit of g(x) = sin(x) / x as x approaches 0
Answer
Let’s calculate the limit of g(x) = sin(x) / x as x approaches 0.
1. Direct Substitution:
- If we directly substitute x = 0 into the function, we get sin(0) / 0 = 0/0. This is an indeterminate form, meaning we can’t determine the limit just by plugging in the value.
2. Applying L’Hôpital’s Rule:
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L’Hôpital’s Rule states that if the limit of f(x)/g(x) as x approaches a certain value results in an indeterminate form (like 0/0 or ∞/∞), then the limit of f(x)/g(x) is equal to the limit of f’(x)/g’(x), provided the latter limit exists. Here, f’(x) and g’(x) are the derivatives of f(x) and g(x) respectively.
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Let’s apply L’Hôpital’s Rule to our problem:
- f(x) = sin(x) => f’(x) = cos(x)
- g(x) = x => g’(x) = 1
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Now we need to find the limit of f’(x)/g’(x) = cos(x) / 1 as x approaches 0.
3. Evaluating the Limit:
- lim_(x->0) cos(x) / 1 = cos(0) / 1 = 1 / 1 = 1
Therefore, the limit of g(x) = sin(x) / x as x approaches 0 is 1.