Evaluate the following limits: _ x 0 [ [3] 1+x -1+x x ] - Vidyadip

Question

Evaluate the following limits:
$\lim _{x \rightarrow 0}\left[\frac{\sqrt[3]{1+x}-\sqrt{1+x}}{x}\right]$

✓

Answer

$ \lim _{x \rightarrow 0} \frac{\sqrt[3]{1+x}-\sqrt{1+x}}{x}$
$=\lim _{x \rightarrow 0} \frac{(1+x)^{\frac{1}{3}}-(1+x)^{\frac{1}{2}}}{x} $
Put $1+x=y$
As $x \rightarrow 0, y \rightarrow 1$
$ \lim _{x \rightarrow 0} \frac{(1+x)^{\frac{1}{3}}-(1+x)^{\frac{1}{2}}}{x}$
$=\lim _{y \rightarrow 1} \frac{y^{\frac{1}{3}}-y^{\frac{1}{2}}}{y-1}$
$=\lim _{y \rightarrow 1} \frac{\left(y^{\frac{1}{3}}-1\right)-\left(y^{\frac{1}{2}}-1\right)}{y-1}$
$=\lim _{y \rightarrow 1}\left(\frac{y^{\frac{1}{3}}-1}{y-1}-\frac{y^{\frac{1}{2}}-1}{y-1}\right)$
$=\lim _{y \rightarrow 1} \frac{y^{\frac{1}{3}}-1^{\frac{1}{3}}}{y-1}-\lim _{y \rightarrow 1} \frac{y^{\frac{1}{2}}-1^{\frac{1}{2}}}{y-1}$
$=\frac{1}{3}(1)^{\frac{-2}{3}}-\frac{1}{2}(1)^{\frac{-1}{2}} \quad \cdots\left[\because \lim _{x \rightarrow a} \frac{x^n-\mathrm{a}^n}{x-\mathrm{a}}=\mathrm{n} \cdot \mathrm{a}^{\mathrm{n}-1}\right]$
$=\frac{1}{3}-\frac{1}{2}$
$=\frac{2-3}{6}$
$=-\frac{1}{6}$

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 "Solve the Following Question.(3 Marks)" from Limits→Limits — all question types→Maths STD 11 Science question papers→Maharashtra Board STD 11 Science question papers→Maharashtra Board question paper generator→

Similar questions

In the following expansions, find the indicated coefficients.

$x^{-3}$ in $\left(x-\frac{1}{2 x}\right)^5$

If $\text{a}^2,\ \text{b}^2,\ \text{c}^2$ are in A.P., prove that $\frac{\text{a}}{\text{b}+\text{c}},\frac{\text{b}}{\text{c}+\text{a}},\frac{\text{c}}{\text{a}+\text{b}}$ are in A.P.

A piece of equipment cost a certain factory ₹ 600,000. If it depreciates in value, 15% the first, 13.5% the next year, 12% the third year, and so on. What will be its value at the end of 10 years, all percentages applying to the original cost?

Solve the following system of equations in R.
$1\leq|\text{x}-2|\leq3$

In an examination, a question paper consists of 12 questions divided into two parts i.e. Part I and Part II, containing 5 and 7 questions, respectively. A student is required to attempt 8 questions in all, selecting at least 3 from each part. In how many ways can a student select the questions?

Differentiate the following function with respect to x:$(\text{ax}+\text{b})^\text{n}(\text{cx}+\text{d})^\text{m}$

Differentiate the following function with respect to $(\text{x})$:$\frac{2\text{x}^2+\text{3x}+4}{\text{x}}$

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow0}\frac{\text{x}\cos\text{x}+2\sin\text{x}}{\text{x}^2+\tan\text{x}}$

Evaluate $\lim\limits_{\text{x}\rightarrow2}{\text{f(x)}}$ (if it exist), where $\text{f(x)}=\begin{cases}\text{x}-[\text{x}],&\text{x}<2\\4 ,& \text{x} = 2\\3\text{x}-5, & \text{x} > 2\end{cases}.$

Show the following quadratic equation by factorization method:
$21x^2 - 28x + 10 = 0$