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Kamila [148]
3 years ago
12

Show why the limit as x approaches 0 (csc(x)-cot(x)) involves an indeterminate form, and then prove that the limit equals 0.

Mathematics
1 answer:
Nadusha1986 [10]3 years ago
5 0

Answer with Step-by-step explanation:

We are given that \lim_{x\rightarrow 0 }(csc(x)-cot(x))

We have to prove that why the limit x approaches 0(csc(x)-cot(x)) involves an indeterminate form and prove that the limit equals to 0.

\lim_{x\rightarrow 0 }(\frac{1}{sinx}-\frac{cosx}{sinx})

Because csc(x)=\frac{1}{sinx} andcot(x)=\frac{cosx}{sinx}

\lim_{x\rightarrow 0 }(\frac{1-cosx}{sinx})

\frac{1-cos0}{sin0}

We know that cos 0=1 and sin 0=0

Substitute the values then we get

\frac{1-1}{0}=\frac{0}{0}

We know that \frac{0}{0} is indeterminate form

Hence, the limit x approaches 0(csc(x)-cot(x)) involves an indeterminate form.

L'hospital rule:Apply this rule and  differentiate numerator and denominator separately when after applying \lim_{x\rightarrow a }we get indeterminate form\frac{0}{0}

Now,using L' hospital rule

\lim_{x\rightarrow 0 }\frac{0+sinx}{cosx}

because \frac{dsinx}{dx}=cosx,\frac{dcosx}{dx}=-sinx}

Now, we get

\lim_{x\rightarrow 0 }\frac{sinx}{cos x}

\frac{sin0}{cos0}

\frac{0}{1}=0

Hence,\lim_{x\rightarrow 0 }(csc(x)-cot(x))=0

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