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Problem 64 Hard Difficulty

To prove that sine is continuous, we need to show that $ \lim_{x \to a} \sin x = \sin a $ for every real number $ a $. By Exercise 63 an equivalent statement is that $$ \lim_{h \to 0} \sin (a + h) = \sin a $$.

Use (6) to show that this is true.

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Video Transcript

this problem Number sixty four of the Stuart Calculus eighth edition, section two point five. To prove the sinus continuous, we need to show that the limit is expert is a The function sign of X is equal to sign of A for every real number, eh? I exercised sixty three, and a cool and statement is that the limit is a H bridge. Zero of sign of the quantity April's age is equal to sign of a ah use six to show that this is true. So we will be using an identity to confirm that this isn't true. Identities Sign of experts Why is equal to cynics cosign y plus cosign x sanely that we will pry it here directly if we take this limit and sage approaches zero of the function given sign of the quality of those h, we can separate this into sign of a Kosanovic plus AH co sign a V sign of age. And little by little, we will separate the limit out. H approaches zero. This is a sum inside that would limit. We can use a limit law to separate this into, ah, two different limits. So the first limited sign echoes and h the second limit Each approaches zero of co sign a sign h what we will see in this The great limit is that h approaching zero for this term Sign of age Coast to zero since sign of H Senat zero is equal to zero. So we're left with this first limit. And if we approach zero for H, well, we have the first part, which is a sign of a and then we're multiplying this by the limiters h broke Ziro cosign of each. What is this limit? The second limit equal to well as HBO zero cosign of h approaches one. So this limit is actually equal to one. And what were left over with his the limit as they H bridges are a sign of E and in this case, Santa is independent of H. It is a constant value no matter what age approaches this values equal to their This limit is equal to Santa Fe, and what we've done here is what we have confirmed exactly. This limit this definition for the continuity of sign for any value they on this confirms that this is a D true and sign is continuous