Evaluate the following limit: _ x 2 x^2-x-2 x^2-2x+ (x-2) - Vidyadip

Question

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow2}\frac{\text{x}^2-\text{x}-2}{\text{x}^2-2\text{x}+\sin(\text{x}-2)}$

✓

Answer

$\lim\limits_{\text{x}\rightarrow2}\frac{\text{x}^2-\text{x}-2}{\text{x}^2-2\text{x}+\sin(\text{x}-2)}$
$=\lim\limits_{\text{x}\rightarrow2}\frac{(\text{x}-2)(\text{x}+1)}{\text{x}^2-2\text{x}+\sin(\text{x}-2)}$
$=\lim\limits_{\text{x}\rightarrow2}\frac{1}{\frac{\text{x}}{\text{x}+1}+\frac{\sin(\text{x}-2)}{(\text{x}-2)(\text{x}+1)}}$
$=\lim\limits_{\text{x}\rightarrow2}(\text{x}+1)\Bigg(\frac{1}{\text{x}+\frac{\sin(\text{x}-2)}{\text{x}-2}}\Bigg)$
$=\lim\limits_{\text{x}\rightarrow2}(\text{x}+1)\frac{1}{\lim\limits_{\text{x}\rightarrow2}(\text{x})+\lim\limits_{\text{x}\rightarrow2}\frac{\sin(\text{x}-2)}{\text{x}-2}}$
$=(2+1)\times\frac{1}{(2)\lim\limits_{\text{x}\rightarrow2\rightarrow0}\frac{\sin(\text{x}-2)}{(\text{x}-2)}}$ $\Big[\because\lim\limits_{\theta\rightarrow0}\frac{\sin\theta}{\theta}=1\Big]$
$=3\times\frac{1}{2+1}$
$=1$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "5 Marks Questions" from Limits→Limits — all question types→Maths STD 11 Science question papers→CBSE Board STD 11 Science question papers→CBSE Board question paper generator→

Similar questions

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow\frac{\pi}{8}}\frac{\cot4\text{x}-\cos4\text{x}}{(\pi-8\text{x})^3}$

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{\text{a}\sin\text{x}-\text{x}\sin\text{a}}{\text{ax}^2-\text{xa}^2}$

The sides of a square are $x = 6, x = 9, y = 3$ and $y = 6$. Find the equation of a circle drawn on the diagonal of the square as its diameter.

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow{\frac{\pi}{6}}}\frac{\cot^2\text{x}-3}{\text{cosec x}-2}$

If the $4^{th}, 10^{th}$ and $16^{th}$ terms of a G.P. are $x, y$ and $z$ respectiveiy. Prove that $x, y, z$ are in G.P.

A person observes the angle of elevation of the peak of a hill from a station to be $\alpha.$ He walks c metres along a slope inclined at an angle $\beta$ and finds the angle of elevation of the peak of the hill to be $\gamma.$ Show that the height of the peak above the ground is $\frac{\text{c}\sin\alpha\sin(\gamma-\beta)}{(\sin\gamma-\alpha)}.$

Differentiate the following from first principle$\tan(\text{2x}+1)$

Evaluate the following limit:
Evaluate: $\lim\limits_{\text{n}\rightarrow\infty}\frac{1^4+2^4+3^4+\ \cdots+\text{n}^4}{\text{n}^5}-\lim\limits_{\text{n}\rightarrow\infty}\frac{1^3+2^3+\ \cdots+\text{n}^3}{\text{n}^5}$

How many litres of water will have to be added to 1125 litres of the 45% solution of acid so that the resulting mixture will contain more than 25% but less than 30% acid content?

$\frac{\text{c}}{\text{a}-\text{b}}=\frac{\tan\big(\frac{\text{A}}{2}\big)+\tan\big(\frac{\text{B}}{2}\big)}{\tan\big(\frac{\text{A}}{2}\big)-\tan\big(\frac{\text{B}}{2}\big)}$