Question

${lim}_{x\to 0}\left(\frac{\sec ax-\sec bx}{x^2}\right)=$

• A. $\frac{a^2-b^2}{2}$
• B. $\frac{b^2-a^2}{2}$
• C. $0$
• D. $1$

Answer 1

Answer: (A)

$\frac{a^2-b^2}{2}$

$\underset{x\to 0}{lim}\frac{\sec ax-\sec bx}{x^2}\left(\frac{0}{0}\right)$ form

$=\underset{x\to 0}{lim}\frac{a\sec ax\tan ax-b\sec bx\tan bx}{2x}$

$=\underset{x\to 0}{lim}\frac{a\sec ax\tan ax}{2x}-\underset{x\to 0}{lim}\frac{b\sec bx\tan bx}{2x}$

$=a\times 1\times \frac{a}{2}-b\times 1\times \frac{b}{2}$

$=\frac{a^2}{2}-\frac{b^2}{2}$

$=\frac{a^2-b^2}{2}$

$L=\frac{a^2-b^2}{2}$