If the function f(x) = ${\begin{matrix} \frac{72^{x} - 9^{x} - 8^{x} + 1}{\sqrt{2} - \sqrt{1 + \cos x}}, x \neq 0 \\ a \log_{e} 2 \log_{e} 3, x = 0 \end{matrix}$ is continuous at x = 0, then the value of a² is equal to

Question

Question

Download Solution PDF

If the function f(x) = ${\begin{matrix} \frac{72^{x} - 9^{x} - 8^{x} + 1}{\sqrt{2} - \sqrt{1 + \cos x}}, x \neq 0 \\ a \log_{e} 2 \log_{e} 3, x = 0 \end{matrix}$ is continuous at x = 0, then the value of a² is equal to

968
1152
746
1250

Answer 1

Concept -

$lim_{n \to 0} (\frac{8^{x} - 1}{x}) = l n 2$

Explanation -

$lim_{x \to 0} f (x)$ = a ℓn2 ℓn3

$lim_{x \to 0} \frac{72^{x} - 9^{x} - 8^{x} + 1}{\sqrt{2} - \sqrt{1 + \cos x}} = lim_{x \to 0} \frac{(8^{x} - 1) (9^{x} - 1)}{\sqrt{2} - \sqrt{1 + \cos x}}$

= $lim_{n \to 0} (\frac{8^{x} - 1}{x}) (\frac{9^{x} - 1}{x}) (\frac{x^{2}}{1 - \cos x}) (\sqrt{2} + \sqrt{1 + \cos x})$

= ℓn 8 × ℓn 9 × 2 × 2√{2} = 24√{2} ℓn2 ℓn3

Given that function is continuous at x = 0

∴ a = 24√{2} ⇒ a2 = 576 × 2 = 1152

Download Solution PDF

Question

If the function f(x) = ${\begin{matrix} \frac{72^{x} - 9^{x} - 8^{x} + 1}{\sqrt{2} - \sqrt{1 + \cos x}}, x \neq 0 \\ a \log_{e} 2 \log_{e} 3, x = 0 \end{matrix}$ is continuous at x = 0, then the value of a² is equal to

Answer (Detailed Solution Below)

Detailed Solution

More Continuity of a function Questions

More Limit and Continuity Questions

More Mathematics Questions

Question

If the function f(x) = {72x−9x−8x+12−1+cos⁡x,x≠0aloge⁡2loge⁡3,x=0 is continuous at x = 0, then the value of a2 is equal to

Answer (Detailed Solution Below)

Detailed Solution

More Continuity of a function Questions

More Limit and Continuity Questions

More Mathematics Questions

If the function f(x) = ${\begin{matrix} \frac{72^{x} - 9^{x} - 8^{x} + 1}{\sqrt{2} - \sqrt{1 + \cos x}}, x \neq 0 \\ a \log_{e} 2 \log_{e} 3, x = 0 \end{matrix}$ is continuous at x = 0, then the value of a² is equal to