Question

Let $f\left(x\right)=3x^{10}-7x^8+5x^6-21x^3+3x^2-7.\text{Then}\underset{h\to 0}{lim}\frac{f(1-h)-f\left(1\right)}{h^3+3h}$

• A. does not exist
• B. is $\frac{50}{3}$
• C. is $\frac{53}{3}$
• D. is $\frac{22}{3}$

Answer 1

The correct option is C is $\frac{53}{3}$

Let,
$L=\underset{h\to 0}{lim}\frac{f(1-h)-f\left(1\right)}{h^3+3h}\cdots \left(\frac{0}{0}\text{form}\right)$
We know that,
$f^{\prime }\left(1\right)=\underset{h\to 0}{lim}\frac{f(1-h)-f\left(1\right)}{-h}$
Using this,
$L=f^{\prime }\left(1\right)\underset{h\to 0}{lim}\frac{-h}{h^3+3h}\Rightarrow L=-53\times \frac{-1}{3}=\frac{53}{3}$