Prove that the function f(x) = 5x – 3 is continuous at x = 0, at … - Vidyadip

Question

Prove that the function f(x) = 5x – 3 is continuous at x = 0, at x = – 3 and at x = 5.

✓

Answer

Here f(x) = 5x - 3

$\ \ \ \text{Lt}\ \ \ \ \ \ \text{f(x)}\\ \text{x} \rightarrow 0$ $= \ \ \ \text{Lt}\ \ \ \ \ \ (5\text{x}-3) = 5(0) - 3 = 0 - 3 = -3\\ \ \ \ \ \text{x} \rightarrow 0$

Now f is defined at x = 0

and f(0) = 5(0) - 3 = 0 - 3 = -3

$\therefore \ \ \text{Lt}\ \ \ \ \ \ \ \ \ \ \ \text{f(x)} = \text{f}(0) = -3\\ \ \ \ \text{x}\rightarrow0$

$\therefore$ f is continous at x = 0

$\ \ \ \text{Lt}\ \ \ \ \ \ \text{f(x)}\\ \text{x} \rightarrow -3$$= \ \ \ \text{Lt}\ \ \ \ \ \ (5\text{x}-3) = 5(-3) - 3 = -15- 3 = -18\\ \ \ \ \ \text{x} \rightarrow -3$

Now f is defined at x = -3

and f(-3) = 5(-3) - 3 = -15 - 3 = -18

$\therefore \ \ \text{Lt}\ \ \ \ \ \ \ \ \ \ \ \text{f(x)} = \text{f}(-3) = -18\\ \ \ \ \text{x}\rightarrow-3$

$\therefore$ f is continous at x = -3

$\ \ \ \text{Lt}\ \ \ \ \ \ \text{f(x)}\\ \text{x} \rightarrow 5$$= \ \ \ \text{Lt}\ \ \ \ \ \ (5\text{x}-3) = 5(5) - 3 = 25- 3 = 22\\ \ \ \ \ \text{x} \rightarrow 5$

Now f is defined at x = 5

and f(5) = 5(5) - 3 = 25 - 3 = 22

$\therefore \ \ \text{Lt}\ \ \ \ \ \ \ \ \ \ \ \text{f(x)} = \text{f}(5) = 22\\ \ \ \ \text{x}\rightarrow5$

$\therefore$ f is continous at x = 5

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 "3 Marks" from CONTINUITY AND DIFFERENTIABILITY→CONTINUITY AND DIFFERENTIABILITY — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Solve: $\cos\Big\{2\sin^{-1}\{-\text{x}\}\Big\}=0$

Find which of the binary operations are commutative and which are associative.
Let A = N × N and * be the binary operation on A defined by:
(a, b) * (c, d) = (a + c, b + d)

Determine that value of the constant 'k' so that function $\text{f(x)}=\begin{cases}\frac{\text{kx}}{|\text{x}|},&\text{if }\text{ x}<0\\3,&\text{if }\text{ x}\geq0\end{cases}$ is continuous at x = 0.

If $\tan^{-1}\text{x}+\tan^{-1}\text{y}=\frac{\pi}{4},$ then write the value of x + y + xy.

Let f be any real function and let g be a function given by g(x) = 2x. Prove that gof = f + f.

Find the vector from the origin O to the centroid of the triangle whose vertices are (1, -1, 2), (2, 1, 3) and (-1, 2, -1).

Find the projection of $\vec{\text{b}}+\vec{\text{c}}$ on $\vec{\text{a}},$ where $\vec{\text{a}}=2\hat{\text{i}}-2\hat{\text{j}}+\hat{\text{k}},\vec{\text{b}}=\hat{\text{i}}+2\hat{\text{j}}-2\hat{\text{k}}$ and $\vec{\text{c}}=2\hat{\text{i}}-\hat{\text{j}}+4\hat{\text{k}}.$

For any two vectors $\vec{\text{a}}$ and $\vec{\text{b}},$ prove that $\big|\vec{\text{a}}\times\vec{\text{b}}\big|^2=\begin{vmatrix}\vec{\text{a}}.\vec{\text{a}}&\vec{\text{a}}.\vec{\text{b}}\\\vec{\text{b}}.\vec{\text{a}}&\vec{\text{b}} .\vec{\text{b}}\end{vmatrix}.$

Three relation R₂is defined in set A = {a, b, c} as follows:
R₂ = {(a, a)}
Find whether or not the relation R₂on A is:

Reflexive.
Symmetric.
Transitive.

Evaluate the following integrals:
$\int\frac{(1+\text{x})^3}{\sqrt{\text{x}}}\text{dx}$