Question

Assertion :$\underset{x\to \infty }{lim}\frac{(x+1{)}^{10}+(x+2{)}^{10}+...+(x+100{)}^{10}}{x^{10}+9^{10}}=100$
If $p\left(x\right)$ and q(x) are polynomials of same degree, then Reason: $\underset{x\to \infty }{lim}\frac{p\left(x\right)}{q\left(x\right)}=\frac{leadingcoefficientsofp\left(x\right)}{leadingcoefficientsofq\left(x\right)}$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

Let $p\left(x\right)=a_0{x}^n+a_0{x}^{n-1}+...+a_n$ and $q\left(x\right)=b_0{x}^n+b_1{x}^{n-1}+...+b_n$ then
$\underset{x\to \infty }{lim}\frac{p\left(x\right)}{q\left(x\right)}=\underset{x\to \infty }{lim}\frac{a_0+a_1/x+a_2/x^2+....+a_n/x^n}{b_0+b_1/x+b_2/x^2+...+b_n/x^n}$
$=\frac{a_0}{b_0}$
$(x+1{)}^{10}+....+(x+100{)}^{10}$ is a polynomial of degree 10 with leading coefficient 100 and $x^{10}+9^{10}$ is a polynomial of degree 10 with leading coefficient 1 no statement 1 follows from statement-2.