Download App

Limits — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsLimits4 Marks
Question
Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow3}\frac{\text{x}^{\text{n}}-3^\text{n}}{\text{x}-\text{3}}=108,$ find the value of n.
✓

Answer

$\lim\limits_{\text{x}\rightarrow3}\frac{\text{x}^{\text{n}}-3^\text{n}}{\text{x}-\text{3}}=108$$\text{L.H.S}=\lim\limits_{\text{x}\rightarrow{3}}\frac{\text{x}^{\text{n}}-3^{\text{n}}}{\text{x}-3}$
Applying the 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, n = n, a = 3
$\Rightarrow{\lim\limits_{\text{x}\rightarrow{3}}}\frac{\text{x}^{\text{n}}-3^\text{n}}{\text{x}-3}=\text{n}(3)^{\text{n}-1}$
It is given that $\text{n}(3)^{\text{n}-1}=108$
$\Rightarrow\text{n}(3)^{\text{n}-1}=2\times2\times3\times3\times3$
$=(2)^2\times(3)^3$
$=4(3)^{4-1}$
$\Rightarrow\text{n}=4$

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.(4 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

Prove by the method of induction, for all n ∈ N.

$(\cos \theta+i \sin \theta)^n=\cos (n \theta)+i \sin (n \theta)$

→
If $\text{z}_1=2-\text{i}, \ \text{z}_2=1+\text{i},$ Find $\Big|\frac{\text{z}_1+\text{z}_2+1}{\text{z}_1-\text{z}_2+\text{i}}\Big|.$
→
If $A+B+C=\frac{3 \pi}{2}$, then $\cos 2 A+\cos 2 B+\cos 2 C=1-4 \sin A$ $\sin B \sin C$
→
How many different words can be formed from the letters of the word 'GANESHPURI'? In how many of these words:
  1. The letter G always occupies the first place?
  2. The letters P and I respectively occupy first and last place?
  3. The vowels are always together?
  4. The vowels always occupy even places?
→
$\text{a}\sin\frac{\text{A}}{2}\sin\Big(\frac{\text{B}-\text{C}}{2}\Big)+\text{b}\sin\frac{\text{B}}{2}\sin\Big(\frac{\text{C}-\text{A}}{2}\Big)+\text{c}\sin\frac{\text{C}}{2}\sin\Big(\frac{\text{A}-\text{B}}{2}\Big)=0.$
→
Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow0}\frac{\sqrt{\text{a}^2+\text{x}^2}-\text{a}}{\text{x}^2}$
→
The sum of first two terms of an infinite G.P. is 5 and each term is three times the sum of the succeeding terms. Find the G.P.
→
Find three numbers in G.P. whose Product is 729 and the sum of their products in pairs is 819.
→
From 4 officers and 8 jawans in how many ways can 6 be chosen:
  1. To include exactly one officer.
  2. To include at least one officer?
→
The ratio of boys to girls in a college is 3 : 2 and 3 girls out of 500 and 2 boys out of 50 of that college are good singers. A good singer is chosen. What is the probability that the chosen singer is a girl?
→