Problem: Lim sin(cos(x))/sec(x) x -> 0 The answer in the book is sin(1).. which is obvious... but why do I not arrive at the same answer doing is this way..... Knowing as x approaches zero, sin(x)/x approaches 1.... ( Sin(cos(x))/cos(x) )(cos^2(x)) = (1)(1) = 1 = / = sin(1)...???
Ok, so since cos(x) goes to 1: [tex]\lim_{x \to 0} \frac{\sin(\cos(x))}{\cos(x)}[/tex] is equivalent to: [tex]\lim_{u \to 1} \frac{\sin(u)}{u}[/tex] We can't apply the sin(_)/_ rule here. The term inside of the sine approaches 1.
[tex]\lim_{x \to 0}\frac{\sin(x)}{x} = 1[/tex] x needs to approach 0. If it approaches any other value (or does not exist), you can't use this identity.