Question

Evaluate: $\underset{x\to 0}{lim}\frac{x({10}^x-x)}{1-\cos x}$

• A. $\log 10$
• B. $2\log 10$
• C. $3\log 10$
• D. $4\log 10$

Answer 1

Answer: (B)

$2\log 10$

Given: $\underset{x\to 0}{lim}\frac{x({10}^x-x)}{1-\cos x}$

$=\underset{x\to 0}{lim}\frac{x({10}^x-1)}{2{\sin }^2\frac{x}{2}}$

$=2\cdot \underset{x\to 0}{lim}\frac{\frac{({10}^x-1)}{x}}{{\left(\frac{\sin \frac{x}{2}}{\frac{x}{2}}\right)}^2}$

$=2\cdot \frac{\log 10}{1^2}=2\log 10$..............As $[\underset{x\to 0}{lim}\frac{(a^x-1)}{x}=\log aand\underset{x\to 0}{lim}\frac{sinx}{x}=1]$

Hence option ${}^{\prime }{B}^{\prime }$ is the answer.