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

Question

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow{\infty}}{\sqrt{\text{x}^2+\text{cx}}-\text{x}}{}$

✓

Answer

$\lim\limits_{\text{x}\rightarrow{\infty}}{\sqrt{\text{x}^2+\text{cx}}-\text{x}}{}$$=\lim\limits_{\text{x}\rightarrow{\infty}}\Bigg(\big(\sqrt{\text{x}^2+\text{cx}}-\text{x}\big)\frac{\big(\sqrt{\text{x}^2+\text{cx}}+\text{x}\big)}{\sqrt{\text{x}^2+\text{cx}}+\text{x}}\Bigg)$
$=\lim\limits_{\text{x}\rightarrow{\infty}}\frac{\big(\text{x}^2+\text{cx}-\text{x}^2\big)}{\sqrt{\text{x}^2+\text{cx}+\text{x}}}$
$=\lim\limits_{\text{x}\rightarrow{\infty}}\frac{\text{cx}}{\sqrt{\text{x}^2+\text{cx}+\text{x}}}$ $\Big[\frac{\infty}{\infty}\text{ from}\Big]$$=\lim\limits_{\text{x}\rightarrow{\infty}}\frac{\text{c}}{\sqrt{1+\frac{\text{c}}{\text{x}}+1}}$
$=\frac{\text{c}}{1+1}=\frac{\text{c}}{2}$

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:
$\lim\limits_{\text{x}\rightarrow1}\frac{\sqrt{3+\text{x}}-\sqrt{5-\text{x}}}{\text{x}^2-1}$

Prove that:
$\frac{\cos\text{A}+\cos\text{B}}{\cos\text{B}-\cos\text{A}}=\cot\Big(\frac{\text{A}-\text{B}}{2}\Big)\cot\Big(\frac{\text{A}-\text{B}}{2}\Big)$

There are $10$ professors and $20$ students out of whom a committee of $2$ professors and $3$ students is to be formed. Find the number of ways in which this can be done. Further find in how many of these committees:

A particular professor is included.
A particular student is included.
A particular student is excluded.

If $\text{a},\ \text{b},\ \text{c}$ are in A.P., then show that:
$\text{a}^2(\text{b}+\text{c}),\ \text{b}^2(\text{c}+\text{a}),\ \text{c}^2(\text{a}+\text{b})$ are also in A.P.

Let r and n be positive integers such that 1 < r < n. Then prove the following:
$\frac{{^\text{n}}\text{C}_{\text{r}}}{{^\text{n}}\text{C}_{\text{r}-1}}=\frac{\text{n}-\text{r}+1}{\text{r}}$

Find the equations to the straight lines which pass through the origin and are inclined at an angle of 75° to the straight line $\text{x}+\text{y}+\sqrt{3}(\text{y}-\text{x})=\text{a}.$

If $A + B + C =\pi,$ prove that $\frac{\sin 2 A+\sin 2 B+\sin 2 C}{\sin A+\sin B+\sin C}=8 \sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2}$

Differentiate log sin $x$ from first principles.

Find the equation to the ellipse in the following case:
eccentricity $\text{e}=\frac{1}{2}$ and semi-major axis $= 4$

Use the Principle of Mathematical Induction in the following Exercis.
A sequence $a_1, a_{2,} a_3 ......$ is defined by letting $a_1 = 3$ and $a_k = 7a_{k-1}$ for all natural numbers $\text{k}\geq2.$ Show that an $= 3.7^{n-1}$ for all natural numbers.