If limx-1ln x-2x-a-bln a where a- b- then the value of a-b 2 is

Let L=lim_x→1ln x·π2^x ⇒ L=lim_x→1ln xcosπ2^x (0/0 form) ⇒ L=lim_x→11/x sin(π2^x)×( πln22^x) ⇒ L=lim_x→12^xxsin(π2^x)×πln 2 ⇒ L =2πln 2 ⇒ a=2, b=1 ∴ (a+b)^2 ...

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Standard XII

Mathematics

Single Point Continuity

If limx→1ln x...

Question

If $lim x \to 1 ln x \cdot sec \frac{π}{2^{x}} = \frac{a}{π^{b} ln a}$ where $a, b \in N$ , then the value of ${(a + b)}^{2}$ is

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Solution

Let $L = lim x \to 1 ln x \cdot sec \frac{π}{2^{x}}$
$\Rightarrow L = lim x \to 1 \frac{ln x}{cos \frac{π}{2^{x}}} (0 / 0 form)$
$\Rightarrow L = lim x \to 1 \frac{1 / x}{- sin (\frac{π}{2^{x}}) \times (\frac{- π ln 2}{2^{x}})}$
$\Rightarrow L = lim x \to 1 \frac{2^{x}}{x sin (\frac{π}{2^{x}}) \times π ln 2}$
$\Rightarrow L = \frac{2}{π ln 2} \Rightarrow a = 2, b = 1 ∴ (a + b)^{2} = 9$

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If limx→1lnx⋅secπ2x=aπblna where a,b∈N, then the value of (a+b)2 is

Let L=limx→1lnx⋅secπ2x ⇒L=limx→1lnxcosπ2x (0/0 form) ⇒L=limx→11/x−sin(π2x)×(−πln22x) ⇒L=limx→12xxsin(π2x)×πln2 ⇒L=2πln2⇒a=2,b=1∴(a+b)2=9

If $lim x \to 1 ln x \cdot sec \frac{π}{2^{x}} = \frac{a}{π^{b} ln a}$ where $a, b \in N$ , then the value of ${(a + b)}^{2}$ is