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

Evaluate the following integrals:
$\int\limits^\frac{\pi}{2}_{-\frac{\pi}{2}}\text{x}\cos^2\text{x}\text{ dx}$

Prove that the function f : N → N, defined by f(x) = x² + x + 1, is one-one but not onto.

The plane ax + by = 0 is rotated about its line of intersection with the plane z = 0 through an angle $\alpha.$ Prove that the equation of the plane in its new position is $\text{ax}+\text{by}\pm\bigg(\sqrt{\text{a}^2+\text{b}^2}\tan\alpha\bigg)\text{z}=0.$

Evaluate the following definite integrals:
$\int_{0}^\limits{\frac{\pi}{2}}\sin\text{x }\sin2\text{x}\text{ dx}$

Three machines E₁, E₂, E₃ in a certain factory produce 50%, 25% and 25%, respectively, of the total daily output of electric bulbs. It is known that 4% of the tubes produced one each of the machines E₁and E₂ are defective, and that 5% of those produced on E₃ are defective. If one tube is picked up at random from a day's production, then calculate the probability that it is defective.

If $\big[3\vec{\text{a}}+7\vec{\text{b}}\vec{\text{c}}\vec{\text{d}}\big]=\lambda\big[\vec{\text{a}}\vec{\text{c}}\vec{\text{d}}\big]+\mu\big[\vec{\text{b}}\vec{\text{c}}\vec{\text{d}}\big],$ then find the value of $\lambda+\mu.$

Find the value of x if the area of $\triangle$ is 35 square cms with vertices (x, 4), (2, -6) and (5, 4).

Solve the following differential equation:

$x \cos \text{y dy} = ( xe^{x} \log x + e^{x}) dx$

Discuss the continuity of the function $\text{f(x)}=\begin{cases}\frac{\text{x}}{|\text{x}|},&\text{x}\neq0\\0,&\text{x}=0\end{cases}$

An integer m is said to be related to another integer n if m is a multiple of n.Check if the relation is symmetric, reflexive and transitive.