17Calculus Proof - Pinching Theorem sin(x)/x
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This page contains two proofs of the limit \[ \lim_{\theta \to 0}{\frac{\sin(\theta)}{\theta}} = 1 \] This limit can also be proved using the Pinching Theorem.
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Proof Using Trig |
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Remember that, in general, \(0/0\) is indeterminate. So in this case we can't just say that \(0/0 = 1\) without special techniques. This video uses trig to prove the limit.
prove that |
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\(\displaystyle{ \lim_{\theta \to 0}{\frac{\sin(\theta)}{\theta}} = 1 }\) |
An informal proof is shown in this video.
Khan Academy - Proof of sin(x)/x Limit [18min-4secs]
video by Khan Academy |
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Proof Using L'Hopitals Rule |
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If you know L'Hopitals Rule, it is easy to show the limit holds. However, most of the time, this limit is introduced before learning L'Hopitals Rule. But if you have learned L'Hopitals Rule, you will understand these steps.
\(\displaystyle{ \lim_{\theta \to 0}{\frac{\sin(\theta)}{\theta}} }\) |
Direct substitution shows that we get \(0/0\) which is indeterminate, so we can use L'Hopitals Rule. |
\(\displaystyle{ \lim_{\theta \to 0}{\frac{\cos(\theta)}{1}} }\) |
Trying direct substitution again gives us \(1/1 = 1\). |
\(\displaystyle{ \lim_{\theta \to 0}{\frac{\sin(\theta)}{\theta}} = 1 }\) -qed- |
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