If limx-010x2-cos xx2 - ln-12- then the value of - is

Lim_x→010^x^2 cos xx^2 (00 form) =lim_x→ 010^x^2 1 (cos x 1)x^2 =lim_x→ 010^x^2 1+2sin^2(x2)x^2 =lim_x→ 010^x^2 1x^2+lim_x→ 02sin^2(x2)4×x^24 =ln10+12 ∴α ...

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Byju's Answer

Standard XII

Mathematics

Differentiability

If limx→010x2...

Question

If $lim x \to 0 \frac{10^{x^{2}} - cos x}{x^{2}} = ln α + \frac{1}{2}$ , then the value of $α$ is

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Solution

$lim x \to 0 \frac{10^{x^{2}} - cos x}{x^{2}} (\frac{0}{0} form) = lim x \to 0 \frac{10^{x^{2}} - 1 - (cos x - 1)}{x^{2}} = lim x \to 0 \frac{10^{x^{2}} - 1 + 2 {sin}^{2} (\frac{x}{2})}{x^{2}} = lim x \to 0 \frac{10^{x^{2}} - 1}{x^{2}} + lim x \to 0 \frac{2 {sin}^{2} (\frac{x}{2})}{4 \times \frac{x^{2}}{4}} = ln 10 + \frac{1}{2} ∴ α = 10$

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If limx→010x2−cosxx2=lnα+12, then the value of α is

limx→010x2−cosxx2 (00 form)=limx→010x2−1−(cosx−1)x2=limx→010x2−1+2sin2(x2)x2=limx→010x2−1x2+limx→02sin2(x2)4×x24=ln10+12∴α=10

If $lim x \to 0 \frac{10^{x^{2}} - cos x}{x^{2}} = ln α + \frac{1}{2}$ , then the value of $α$ is