Question
If given function is continuous at $x=1$, then find $a$ and $b. f(x)=\left\{\begin{array}{cl} 3 a x+b, & \text { if } x>1 \\ 11, & \text { if } x=1 \\ 5 a x-2 b, & \text { if } x<1 \end{array}\right.$
✓
Answer
Given function $f(x)=\left\{\begin{array}{cl} 3 a x+b, & \text { if } x>1 \\ 11, & \text { if } x=1 \\ 5 a x-2 b, & \text { if } x<1\end{array}\right.$
value of $\text{L.H.L}.$
$\lim _{x \rightarrow 1^{-}} f(x) =\lim _{x \rightarrow 1^{-}}[5 a x-2 b]$
$ =\lim _{h \rightarrow 0}[5 a(1-h)-2 b]$
$ =\lim _{h \rightarrow 0}[5 a-2 b-5 a h]$
$ =5 a-2 b$
$\therefore \lim _{x \rightarrow 1^{-}} f(x) =5 a-2 b .$
value of $\text{R.H.L}.$
$=\lim _{x \rightarrow 1^{+}} f(x)=\lim _{x \rightarrow 1^{+}}(3 a x+b)$
$ =\lim _{h \rightarrow 0}[3 a(1+h)+b]$
$ =\lim _{h \rightarrow 0}(3 a+b+3 a h)=3 a+b$
$\therefore \lim _{x \rightarrow l^{+}} =3 a+b$
and $ f(1) =11$
because function is continuous at $x=1$
$f(1) =\lim _{h \rightarrow 0} f(1+h)=\lim _{h \rightarrow 0} f(1-h)$
$\Rightarrow 11=3 a+b=5 a-2 b$
value of $\text{L.H.L}.$
$\lim _{x \rightarrow 1^{-}} f(x) =\lim _{x \rightarrow 1^{-}}[5 a x-2 b]$
$ =\lim _{h \rightarrow 0}[5 a(1-h)-2 b]$
$ =\lim _{h \rightarrow 0}[5 a-2 b-5 a h]$
$ =5 a-2 b$
$\therefore \lim _{x \rightarrow 1^{-}} f(x) =5 a-2 b .$
value of $\text{R.H.L}.$
$=\lim _{x \rightarrow 1^{+}} f(x)=\lim _{x \rightarrow 1^{+}}(3 a x+b)$
$ =\lim _{h \rightarrow 0}[3 a(1+h)+b]$
$ =\lim _{h \rightarrow 0}(3 a+b+3 a h)=3 a+b$
$\therefore \lim _{x \rightarrow l^{+}} =3 a+b$
and $ f(1) =11$
because function is continuous at $x=1$
$f(1) =\lim _{h \rightarrow 0} f(1+h)=\lim _{h \rightarrow 0} f(1-h)$
$\Rightarrow 11=3 a+b=5 a-2 b$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
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.
→'*' on Q defined by $\text{a}\ ^*\ \text{b}=\frac{\text{ab}}{4}$ for all a, b ∈ Q.
The cost of $4 \ kg$ onion, $3 \ kg$ wheat and $2 \ kg$ rice is $Rs \ 60$. The cost of $2 \ kg$ onion, $4 \ kg$ wheat and $6 \ kg$ rice is $Rs \ 90$. The cost of $6 \ kg$ onion $2 \ kg$ wheat and $3 \ kg$ rice is Rs $70$. Find cost of each item per kg by matrix method.
→Find the equation of a curve passing through (2, 1) if the slope of the tangent to the curve at any points (x, y) is $\frac{\text{x}^2+\text{y}^2}{2\text{xy}}.$
→If $\text{x}=10(\text{t}-\sin\text{t}),\text{y}=12(1-\cos\text{t}),$ find $\frac{\text{dy}}{\text{dx}}.$
→A wholesale dealer deals in two kinds, A and B (say) of mixture of nuts. Each kg of mixture A contains 60 grams of almonds, 30 grams of cashew nuts and 30 grams of hazel nuts. Each kg of mixture B contains 30 grams of almonds, 60 grams of cashew nuts and 180 grams of hazel nuts. The remainder of both mixtures is per nuts. The dealer is contemplating to use mixtures A and B to make a bag which will contain at least 240 grams of almonds, 300 grams of cashew nuts and 540 grams of hazel nuts. Mixture A costs Rs. 8 per kg. and mixture B costs Rs. 12 per kg. Assuming that mixtures A and B are uniform, use graphical method to determine the number of kg. of each mixture which he should use to minimise the cost of the bag.
Find the particular solution of the following differential equation;
$\frac{\text{dx}}{\text{dy}} = 1 + x^2 + y^2 + x^2y^2, $ given that $y = 1$ when $x = 0$.
→$\frac{\text{dx}}{\text{dy}} = 1 + x^2 + y^2 + x^2y^2, $ given that $y = 1$ when $x = 0$.
Show that $\text{f}(\text{x})=\frac{1}{1+\text{x}^2}$ is neither increasing nor decreasing on R.
→Find the particular solution of the differential equation$\text{e}^{x}\sqrt{1 - \text{y}^{2}} \text{ dx} + \frac{\text{y}}{\text{x}}\text{dy } = 0 $ given that y= 1 when x=0.
→Find the equation of the tangent line to the curve $y = x^2 - 2x + 7$ which is perpendicular to the line $5y - 15x = 13.$
→ Show that the points A, B, C with position vectors $2\hat{\text{i}} - \hat{\text{j}} + \hat{\text{k}}, \hat{\text{i}} - 3\hat{\text{j}} - 5\hat{\text{k}} \text{ and } 3\hat{\text{i}} - 4\hat{\text{j}} - 4\hat{\text{k}}$ respectively, are the vertices of a right-angled triangle. Hence find the area of the triangle.
→