Download App

Limits — MATHS STD 11 Science — Question

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

Answer

$\lim\limits_{\text{x}\rightarrow0}\frac{\sqrt{\text{a}^2+\text{x}^2}-\text{a}}{\text{x}^2}$ $=\lim\limits_{\text{x}\rightarrow0}\frac{\big(\sqrt{\text{a}^2+\text{x}^2}-\text{a}\big)}{\text{x}^2}\times\frac{\big(\sqrt{\text{a}^2+\text{x}^2}+\text{a}\big)}{\big(\sqrt{\text{a}^2+\text{x}^2}+\text{a}\big)}$ $=\lim\limits_{\text{x}\rightarrow0}\frac{\big(\text{a}^2+\text{x}^2-\text{a}^2\big)}{\text{x}^2\sqrt{\text{a}^2+\text{x}^2}+\text{a}}$ $=\lim\limits_{\text{x}\rightarrow0}\frac{\text{x}^2}{\text{x}^2\big(\sqrt{\text{a}^2+\text{x}^2}+\text{a}\big)}$ $=\lim\limits_{\text{x}\rightarrow0}\frac{1}{\sqrt{\text{a}^2+\text{x}^2+\text{a}}}$ $=\frac{1}{\text{a}+\text{a}}$ $=\frac{1}{2\text{a}}$

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

Prove the following by using the principle of mathematical induction for all n ∈ N:$\frac{1}{2.5}+\frac{1}{5.8}+\frac{1}{8.11}+...+\frac{1}{(3\text{n}-1)(3\text{n}+2)}=\frac{\text{n}}{(6\text{n}+4)}.$
→
If f, g and h are real functions defined by $\text{f(x)}=\sqrt{\text{x}+1},\text{ g(x)}=\frac{1}{\text{x}}$ and $h(x) = 2x^2 - 3$, find the values of $(2f + g - h)(1)$ and $(2f + g - h) (0)$.
→
Evaluate the following limit: $\lim\limits_{\text{x}\rightarrow-\infty}\big(\sqrt{4\text{x}^2-7\text{x}}+2\text{x}\big)$
→
Prove the following by the principle of mathematical induction: $1.3 + 3.5 + 5.7 + ... + (2\text{n} - 1)(2\text{n} + 1)= \frac{\text{n}(4\text{n}^2+6\text{n}-1)}{3}$
→
Solve the following equation: $\sqrt{3}\text{x}^2-\sqrt{2}\text{x}+3\sqrt{3}=0.$
→
A class consists of 10 boys and 8 girls. Three students are selected at random. What is the probability that the selected group has
  1. All boys?
  2. All girls?
  3. 1 boys and 2 girls?
  4. At least one girl?
  5. At most one girl?
→
Sum the following series to n terms: $2 + 5 + 10 + 17 + 26 + .....$
→
Find the eccentricity, coordinates of the foci, equations of directrices and lenght of the latus-rectum of the hyperbola $2\text{x}^{2}-3\text{y}^{2}=5.$
→
If the lines 3x - 4y + 4 = 0 and 6x - 8y - 7 = 0 are tangents to a circle, then find the radius of the circle.
→
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 $a_n = 3.7^{n-1}$ for all $\text{n}\in\text{N}.$
→