Question
Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow3}\frac{\text{x}^2-\text{x}-6}{\text{x}^3+3\text{x}^2+\text{x}-3}$
$\lim\limits_{\text{x}\rightarrow3}\frac{\text{x}^2-\text{x}-6}{\text{x}^3+3\text{x}^2+\text{x}-3}$
✓
Answer
$\lim\limits_{\text{x}\rightarrow3}\frac{\text{x}^2-\text{x}-6}{\text{x}^3+3\text{x}^2+\text{x}-3}$Now $\text{x}^2-\text{x}-6$
$=\text{x}^2-3\text{x}+2\text{x}-6$
$=\text{x}(\text{x}-3)+2(\text{x}-3)$
$=(\text{x}+2)(\text{x}-3)\ \cdots(\text{i})$
Dividing $\text{x}^3-3\text{x}^2+\text{x}-3\text{ by }(\text{x}-3), \text{ we get}$
Thus (x - 3) is a factor of $\text{x}^3-3\text{x}^2+\text{x}-3\ \cdots(\text{ii})$
Substituting (i) and (ii) in the given expression
$=\lim\limits_{\text{x}\rightarrow3}\frac{(\text{x}+2)(\text{x}-3)}{\big(\text{x}^2+1\big)(\text{x}-3)}$
$=\frac{\text{x}+2}{\text{x}^2+1}=\frac{3+2}{9+1}=\frac{5}{10}$
$=\frac12$
$=\text{x}^2-3\text{x}+2\text{x}-6$
$=\text{x}(\text{x}-3)+2(\text{x}-3)$
$=(\text{x}+2)(\text{x}-3)\ \cdots(\text{i})$
Dividing $\text{x}^3-3\text{x}^2+\text{x}-3\text{ by }(\text{x}-3), \text{ we get}$
Thus (x - 3) is a factor of $\text{x}^3-3\text{x}^2+\text{x}-3\ \cdots(\text{ii})$
Substituting (i) and (ii) in the given expression
$=\lim\limits_{\text{x}\rightarrow3}\frac{(\text{x}+2)(\text{x}-3)}{\big(\text{x}^2+1\big)(\text{x}-3)}$
$=\frac{\text{x}+2}{\text{x}^2+1}=\frac{3+2}{9+1}=\frac{5}{10}$
$=\frac12$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
Differentiate the following functions with respect to x:$\frac{1}{\text{ax}^2+\text{bx}+\text{c}}$
→Prove that:$\sin^2\text{B}=\sin^2\text{A}+\sin^2\text{(A}-\text{B)}-2\sin\text{A}\cos\text{B}\sin\text{(A}-\text{B)} $
→Prove that:
$\cos20^\circ\cos40^\circ\cos80^\circ=\frac{1}{8}$
→$\cos20^\circ\cos40^\circ\cos80^\circ=\frac{1}{8}$
If arg(z - 1) = arg(z + 3i) then find x - 1, y where z = x + iy.
If $\frac{\text{z}-1}{\text{z}+1}$ is purely imaginary number (z ≠ -1), then find the value of |z|.
→Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow2}\frac{\sqrt{3+\text{x}}-1}{2-\text{x}}$
→$\lim\limits_{\text{x}\rightarrow2}\frac{\sqrt{3+\text{x}}-1}{2-\text{x}}$
Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow2}\frac{\text{x}^3+3\text{x}^2-9\text{x}-2}{\text{x}^3-\text{x}-6}$
→$\lim\limits_{\text{x}\rightarrow2}\frac{\text{x}^3+3\text{x}^2-9\text{x}-2}{\text{x}^3-\text{x}-6}$
Find the equation of the straight lines passing through the following pair of points:
$\Big(\text{at}_1,\frac{\text{a}}{\text{t}_1}\Big)$ and $\Big(\text{at}_2,\frac{\text{a}}{\text{t}_2}\Big)$
→$\Big(\text{at}_1,\frac{\text{a}}{\text{t}_1}\Big)$ and $\Big(\text{at}_2,\frac{\text{a}}{\text{t}_2}\Big)$
In any $\triangle A B C$, if $a^2, b^2, c^2$ are in A.P., prove that $\cot A, \cot B$ and $\cot C$ are also in A.P.
→In a $\triangle\text{ABC,}$ if $\sin^2\text{A}+\sin^2\text{B}=\sin^2\text{C},$ show that the triangle is right angled.
→