Download App

Continuity — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsContinuity5 Marks
Question
Examine the continuity of the function
$\text{f}\text{(x)}=\begin{cases}3\text{x}-2 &, \text{ if x} \leq 0\\\text{x}+1 &, \text{ if x} > 0\end{cases}\text{at x}=0$
Also sketch the graph of this function.
✓

Answer

Given function is, $\text{f}\text{(x)}=\begin{cases}3\text{x}-2 &, \text{ if x} \leq 0\\\text{x}+1 &, \text{ if x} > 0\end{cases}\text{at x}=0\ ...(\text{ii})$ We need to check whether f(x) is continuous at x = 0 or not. For this we need to check L.H.L, R.H.L and value of function at x = 0 Clearly, f(0) = 3*0 - 2 = -2 [from equation ii] $\text{L.H.L}=​​\lim\limits_{\text{h} \rightarrow 0}(0-\text{h})=\lim\limits_{\text{h} \rightarrow 0}\text{f}(-\text{h})$ $=\lim\limits_{\text{h} \rightarrow 0}\big\{3(-\text{h})-2\big\}=-2$ $\text{R.H.L}=\lim\limits_{\text{h} \rightarrow 0}\text{f}(0+\text{h})=\lim\limits_{\text{h} \rightarrow 0}\text{f(h)}$ $=\lim\limits_{\text{h} \rightarrow 0}\big\{\text{h}+1\big\}=0+1=1$ As, $\text{L.H.L}\neq\text{R.H.L}$ $\therefore\ \text{f(x)}$ is discontinuous at x = 0 This can also be proved by plotting f(x) on cartesian plane. For x >0 ,we need to plot y = x + 1 put y = 0, we get x = -1 and for second point we put x = 0 and thus get y = 1 Two points are enough to plot the straight line. Two coordinates are (-1, 0) and (0, 1) For $\text{x}\leq0,$ we need to plot y = 3x - 2 put x = 0 then y = -2 On putting y = 0 we get $\text{x}=\frac{2}{3}$ Two coordinates are (0, -2) and $\Big(\frac{2}{3},0\Big)$ It can be seen from graph that there is breakage in curve at (0, 0) Thus, it is discontinuous at x = 0

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.(5 Marks)" from Continuity→Continuity — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

Compute the adjoint of the following matrices:$\begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{bmatrix}$
Verify that (adjoint A)A = |A|I = A (adjoint A) for the above matrices.
→
Find the second order derivatives of the following functions:$\log(\sin\text{x})$
→
On Z, the set of all integers, a binary operation * is defined by a * b = a + 3b - 4. Prove that * is neither commutative nor associative on Z.
→
Find the area of the figure bounded by the curves y = |x - 1| and y = 3 - |x|.
→
Two cards are drawn simultaneously from a pack of 52 cards. Compute the mean and standard deviation of the number of kings.
→
Differentiate the following functions with respect to x:
$\big(\sin^{-1}\text{x}^4\big)^4$
→
Evaluvate the following intregals
$\int\frac{2\text{x}+1}{\sqrt{\text{x}^2+4\text{x}+3}}\text{dx}$
→
Find the equation of the plane determined by the intersection of the lines $\frac{\text{x}+3}{3}=\frac{\text{y}}{-2}=\frac{\text{z}-7}{6}$ and $\frac{\text{x}+6}{1}=\frac{\text{y}+5}{-3}=\frac{\text{z}-1}{2}$
→
Check the commutativity and associativity of the following binary operations:
'*' on Q defined by $\text{a}\ ^*\ \text{b}=\frac{\text{ab}}{4}$ for all a, b ∈ Q.
→
Maximum Z = 3x + 4y Subject to$\text{x}+\text{y}\leq30000$
$\text{y}\leq12000$
$\text{x}\geq6000$
$\text{x}\geq\text{y}$
$\text{x},\text{y}\geq0$
→