Download App

Limits — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSLimits4 Marks
Question
Evaluate the following limit: $\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{\text{x}^{\frac{5}{7}}-\text{a}^{\frac{5}{7}}}{\text{x}^{\frac{2}{7}}-\text{a}^{\frac{2}{7}}}$
✓

Answer

$\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{\text{x}^{\frac{5}{7}}-\text{a}^{\frac{5}{7}}}{\text{x}^{\frac{2}{7}}-\text{a}^{\frac{2}{7}}}$$\Rightarrow\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{\frac{\text{x}^{\frac{5}{7}}-\text{a}^{\frac{5}{7}}}{\text{x}-\text{a}}}{\frac{\text{x}^{\frac{2}{7}}-\text{a}^{\frac{2}{7}}}{\text{x}-\text{a}}}$ [Dividing numrator and denominator by x - a]
$=\frac{\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{\text{x}^{\frac{5}{7}}-\text{a}^{\frac{5}{7}}}{\text{x}-\text{a}}}{\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{\text{x}^{\frac{2}{7}}-\text{a}^{\frac{2}{7}}}{\text{x}-\text{a}}}$
Applying formula $\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{\text{x}^{\text{n}}-\text{a}^\text{n}}{\text{x}-\text{a}}=\text{na}^{\text{n}-1}$
Here, $\text{n}=\frac57$ is numerator and applying $\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{\text{x}^\text{m}-\text{a}^\text{m}}{\text{x}-\text{a}}=\text{ma}^{\text{m}-1}$ in denominator, where $\text{m}=\frac27$
$\Rightarrow\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{\frac{\text{x}^{\frac{5}{7}}-\text{a}^{\frac{5}{7}}}{\text{x}-\text{a}}}{\frac{\text{x}^{\frac{2}{7}}-\text{a}^{\frac{2}{7}}}{\text{x}-\text{a}}}=\frac{\frac{5}{7}\text{a}^{\frac{5}{7}-1}}{\frac{2}{7}(\text{a})^{\frac{2}{7}-1}}$
$=\frac{\frac{5}{7}\text{a}^{\frac{-2}{7}}}{\frac{2}{7}\text{a}^{\frac{-5}{7}}}$
$=\frac{5}{2}\text{a}^{\frac{-2}{7}+\frac{5}{7}}$
$=\frac{5}{2}\text{a}^{\frac{3}{7}}$

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 "(Each question 4 marks)" from Limits→Limits — all question types→MATHS STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

Prove that:
$(\text{A}\cup\text{B})\times\text{C}=(\text{A}\times\text{C})\cup(\text{B}\times\text{C})$
→
The income of a person is ₹ 300,000 in the first year and he receives an increase of ₹ 10000 to his income per year for the next 19 years. Find the total amount, he received in 20 years.
→
Reduce the lines 3x - 4y - 4 = 0 and 2x + 4y - 5 = 0 to the normal form and hence find which line is nearer to the origin.
→
In a village, there are 87 families of which 52 families have at most 2 children. In a rural development programme, 20 families are to be helped chosen for assistance of which at least 18 families must have at most 2 children. In how many ways can the choice be made?
→
If $0\leq\text{x}\leq\pi$ and x lies in the IInd quadrant such that $\sin\text{x}=\frac{1}{4}.$ Find the values of $\cos\frac{\text{x}}{2},\sin\frac{\text{x}}{2}$ and $\tan\frac{\text{x}}{2}$
→
Evaluate the following limit: $\lim\limits_{\text{x}\rightarrow0}\frac{\{\sin(\alpha+\beta)\text{x}+\sin(\alpha-\beta)\text{x}+\sin2\alpha\text{x}\}}{\cos^2\beta\text{x}-\cos^2\alpha\text{x}}$
→
Find the centre, the lengths of the axes, eccentricity, foci of the following ellipse:$\text{x}^2+2\text{y}^2-2\text{x}+12\text{y}+10=0$
→
Five cards are drawn from form a pack of 52 cards. what is the chance that these 5 will contain:
  1. Just one ace
  2. At least one ace.
→
Prove the following by using the principle of mathematical induction for all $n ∈ N: (2n + 7) < (n + 3)^2$.
→
Find the multiplicative inverse of the following complex numbers: $\sqrt{5}-3\text{i}$
→