Understand the problem
True or false: $\lim _{x \rightarrow 0} \frac{\sin x}{\log (1+\tan x)}=1$
Start with hints
Do you really need a hint? Try it first!
"Hint 1"
Observe that the following limit is of the form \(\frac 00\).
Do you remember that we always solve the limits of the form \(\frac 00\) and \(\frac{\infty}{\infty}\) by L’Hospital’s Rule.
Hint 2
- Consider \(f(x)=sinx\) and \(g(x)=log(1+tanx)\)
- Compute \(f ‘(x)=cosx\) and \(g ‘(x) = sec^2(x)/(1+tanx)\)
- Compute the limit of \(f ‘(x)/g ‘(x)\)
Hint 3
- Observe that the \(lim f ‘(x)=1\) and \(lim g ‘(x) =1 \).
- Hence the value of the Limit is \(1\).
- The statement is therefore TRUE.
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