Download App

Limits — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSLimits4 Marks
Question
Evaluate the following limit: $\lim\limits_{\text{h}\rightarrow0}\frac{\text{(a}+\text{h})^2\sin(\text{a}+\text{h})-\text{a}^2\sin\text{a}}{\text{h}}$
✓

Answer

$\lim\limits_{\text{h}\rightarrow0}\frac{\text{(a}+\text{h})^2\sin(\text{a}+\text{h})-\text{a}^2\sin\text{a}}{\text{h}}$ $=\lim\limits_{\text{h}\rightarrow0}\frac{\big(\text{a}^2+\text{h}^2+2\text{ah}\big)\sin(\text{a}+\text{h})-\text{a}^2\sin\text{a}}{\text{h}}$ $=\lim\limits_{\text{h}\rightarrow0}\frac{\text{a}^2\sin(\text{a}+\text{h})+\text{h}^2\sin(\text{a}+\text{h})+2\text{ah}\sin(\text{a}+\text{h})-\text{a}^2\sin\text{a}}{\text{h}}$ $=\lim\limits_{\text{h}\rightarrow0}\Big[\frac{\text{a}^2(\sin(\text{a}+\text{h})-\sin\text{a})}{\text{h}}+\frac{\text{h}^2\sin(\text{a}+\text{h})}{\text{h}(\text{a}+\text{h})}\times(\text{a}+\text{h})+\frac{2\text{ah}}{\text{h}}(\sin(\text{a}+\text{h}))\Big]$ $=\Bigg[\text{a}^2\lim\limits_{\text{h}\rightarrow0}\frac{2\cos\big(\frac{\text{a}+\text{h}+\text{a}}{2}\big)\sin\big(\frac{\text{a}+\text{h}-\text{a}}{2}\big)}{\text{h}}\Bigg]+[0]+2\text{a}\lim\limits_{\text{h}\rightarrow0}\sin(\text{a}+\text{h})$ $=\Big(2\text{a}^2\cos\text{a}\times\frac12\Big)+(2\text{a}\sin\text{a})$ $=\text{a}^2\cos\text{a}+2\text{a}\sin\text{a}$

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

$\cot\text{x}+\cot\Big(\frac{\pi}{3}-\text{x}\Big)+\cot\Big(\frac{\pi}{3}-\text{x}\Big)=3\cot3\text{x}$
→
Let A and B be two stes such that: $\text{n(P)}= 20,$$\text{n(A}\cup\text{B)=42 and n(A}\cap\text{B})=4.$ Find: $\text{n(B} - \text{A)}.$
→
If $z_1$ is a complex number other than -1 such that $|\text{z}|=1$ and $\text{z}_2=\frac{\text{z}_1-1}{\text{z}_1+1},$ then show that the real parts of $z_2$ is zero.
→
If a, b, c are in G.P., prove that: $\frac{\text{a}^2+\text{ab}+\text{b}^2}{\text{bc}+\text{ca}+\text{ab}}=\frac{\text{b}+\text{a}}{\text{c}+\text{b}}$
→
Find the sum of the series whose $n^{th}$ term is: $2n^3 + 3n^2 - 1$
→
calculate the mean deviation about median of the following frequency distribution:
$x_i$
5
7
9
11
13
15
17
$f_i$
2
4
6
8
10
12
8
→
Find the equation to the ellipse in the following case:Foci $(\pm3, 0), a = 4$
→
Solve the following equations: $\sin2\text{x}-\sin4\text{x}+\sin6\text{x}=0$
→
Prove the following by using the principle of mathematical induction for all n ∈ N:$1+\frac{1}{(1+2)}+\frac{1}{(1+2+3)}+...+\frac{1}{(1+2+3+...\text{n})}=\frac{2\text{n}}{(\text{n+1})}.$
→
Evaluate the following limit: $\lim\limits_{\text{x}\rightarrow{\sqrt{2}}}\frac{\sqrt{5+2\text{x}}-\big(\sqrt{2}+1\big)}{\text{x}^2-2}$
→