How to solve trigonometric limit problems
Formulas we use to solve limit problems
Trigonometric limit problems revolve around three formulas:
???\lim_{\theta\to0}\sin{\theta}=0???
???\lim_{\theta\to0}\cos{\theta}=1???
???\lim_{x\to0}\frac{\sin{x}}{x}=1???
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When we solve trigonometric limit problems, our goal is always to reduce the function to a combination of nothing but these three formulas and simple constants.
How to solve trigonometric limit problems with trig limit formulas
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Let’s do an example where we evaluate the trig limit
Example
Evaluate the limit.
???\lim_{x\to0}\frac{1-\cos{x}}{x}???
Since we have exactly two terms in the numerator, we’re actually going to borrow the conjugate method for the first step of this problem.
???\lim_{x\to0}\frac{1-\cos{x}}{x}\cdot\left(\frac{1+\cos{x}}{1+\cos{x}}\right)???
???\lim_{x\to0}\frac{1-\cos^2{x}}{x(1+\cos{x})}???
Applying the identity ???1-\cos^2{x}=\sin^2{x}??? to the numerator gives
???\lim_{x\to0}\frac{\sin^2{x}}{x(1+\cos{x})}???
When we solve trigonometric limit problems, our goal is always to reduce the function to a combination of nothing but these three formulas and simple constants.
Notice now that we can factor out ???(\sin{x})/x???, which is one of our three fundamental formulas.
???\lim_{x\to0}\frac{\sin{x}}{x}\cdot\frac{\sin{x}}{1+\cos{x}}???
???\lim_{x\to0}\frac{\sin{x}}{x}\cdot\lim_{x\to0}\frac{\sin{x}}{1+\cos{x}}???
Since the first limit is one of our three fundamental formulas, we can replace it with its value.
???1\cdot\lim_{x\to0}\frac{\sin{x}}{1+\cos{x}}???
???\lim_{x\to0}\frac{\sin{x}}{1+\cos{x}}???
And now we can evaluate the limit at ???x\to 0??? without making the denominator ???0???.
???\frac{\sin{0}}{1+\cos{0}}???
???\frac{0}{1+1}???
???0???
So the limit is ???0???.
???\lim_{x\to0}\frac{1-\cos{x}}{x}=0???
