Q.Discuss the continuity of the function $f$, where $f$ is defined by $f(x) = \begin{cases} -2, & \text{if } x \le -1 \\ 2x, & \text{if } -1 < x \le 1 \\ 2, & \text{if } x > 1 \end{cases}$

Showing the 12 most recent of 32 on this concept.

1. AP EAPCET 2026Set eng-2026-05-12-FN1 markMCQ
   Q.If a function $f(x) = \begin{cases} \dfrac{2^{\sin^2 x} - \cos bx}{\tan^2 x}, & \text{for } -\dfrac{\pi}{2} < x < 0 \\ 2, & \text{for } x = 0 \\ \dfrac{\sin^2 ax - \sin^2 bx}{1 - \cos x}, & \text{for } 0 < x < \dfrac{\pi}{2} \end{cases}$ is continuous at $x = 0$, then $a^2 + b^2 =$ (A) $9$ (B) $9 - \log 16$ (C) $9 - \log 8$ (D) $1$

   ›Reveal solutionSolution

   Continuity at $x=0$ forces both one-sided limits to equal $f(0)=2$; expanding each piece to second order in $x$ gives two equations in $a^2,b^2$ whose sum is $9-\log16$.

   Concept and Intuition

   A piecewise function is continuous at a boundary point exactly when the left-hand limit, right-hand limit, and the defined value all coincide. Here each branch is a $\frac00$-type expression as $x\to0$, so we expand numerator and denominator to matching (second) order in $x$ and read off the limiting constant.

   Step-by-Step Solution

   1. Right piece, $0<x<\pi/2$: $f(x)=\dfrac{\sin^2(ax)-\sin^2(bx)}{1-\cos x}$. Using $\sin u \approx u$ for small $u$: $\sin^2(ax)-\sin^2(bx)\approx a^2x^2-b^2x^2=(a^2-b^2)x^2$. Using $1-\cos x \approx \dfrac{x^2}{2}$: limit $= \dfrac{(a^2-b^2)x^2}{x^2/2}=2(a^2-b^2)$. Continuity requires this $=f(0)=2$, so $a^2-b^2=1$. — (i)
   2. Left piece, $-\pi/2<x<0$: $f(x)=\dfrac{2^{\sin^2x}-\cos(bx)}{\tan^2x}$. $2^{\sin^2x}=e^{\sin^2x\ln2}\approx 1+x^2\ln2$ (using $\sin^2x\approx x^2$). $\cos(bx)\approx 1-\dfrac{b^2x^2}{2}$. Numerator $\approx x^2\ln2 + \dfrac{b^2x^2}{2} = x^2\left(\ln2+\dfrac{b^2}{2}\right)$. Denominator $\tan^2x \approx x^2$. …

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2. AP EAPCET 2026Set eng-2026-05-14-FN1 markMCQ
   Q.If $f(x) = \begin{cases} \dfrac{\sqrt{2 + \cos x} - 1}{(\pi - x)^2}, & x \neq \pi \\ k, & x = \pi \end{cases}$ is continuous at $x = \pi$, then $k =$ (A) $1$ (B) $\dfrac{1}{2}$ (C) $2$ (D) $\dfrac{1}{4}$

   ›Reveal solutionSolution

   Continuity at $x=\pi$ requires $k$ to equal the limit of the given expression as $x\to\pi$; using a small-angle substitution, that limit is $\dfrac14$.

   Concept and Intuition

   For $f$ to be continuous at $x=\pi$, we need $k = \displaystyle\lim_{x\to\pi} \frac{\sqrt{2+\cos x}-1}{(\pi-x)^2}$. Substituting $x = \pi - h$ (so $h\to 0$ as $x\to\pi$) converts the trig limit into a small-$h$ approximation problem, where standard expansions ($\cos h \approx 1-h^2/2$, $\sqrt{1+t}\approx 1+t/2$) make the limit easy to evaluate.

   Step-by-Step Solution

   1. Let $x = \pi - h$, so as $x\to \pi$, $h\to 0$, and $\pi - x = h$.
   2. $\cos x = \cos(\pi - h) = -\cos h$.
   3. So $2+\cos x = 2 - \cos h$. Using $\cos h \approx 1 - \dfrac{h^2}{2}$ for small $h$: $2-\cos h \approx 2 - 1 + \dfrac{h^2}{2} = 1 + \dfrac{h^2}{2}$.
   4. $\sqrt{2+\cos x} \approx \sqrt{1+\dfrac{h^2}{2}} \approx 1 + \dfrac{h^2}{4}$ (using $\sqrt{1+t}\approx 1+t/2$ with $t=h^2/2$).
   5. So $\sqrt{2+\cos x} - 1 \approx \dfrac{h^2}{4}$.
   6. The denominator is $(\pi-x)^2 = h^2$. …

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3. AP EAPCET 2026Set eng-2026-05-15-AN1 markMCQ
   Q.If $f(x) = \begin{cases} \left(\frac{2+3x^2}{2+5x^2}\right)^{\frac{8}{x^2}+3}, & \text{for } x \neq 0 \\ k, & \text{for } x = 0 \end{cases}$ is a continuous function at $x = 0$, then $k =$ (A) $e^{-2}$ (B) $e^{-4}$ (C) $e^{-8}$ (D) $e^{-16}$

   ›Reveal solutionSolution

   This tests the standard $1^\infty$ exponential limit technique. Taking logs and expanding to first order in $x^2$ gives $k=e^{-8}$.

   Concept and Intuition

   Expressions of the form (base $\to1$)$^{\text{power}\to\infty}$ are evaluated by writing $L=e^{\lim(\text{power})\cdot\log(\text{base})}$ and expanding $\log(1+u)\approx u$ for the small quantity $u=$ base$-1$. The constant added to the exponent (here $+3$) never survives, since it multiplies a term that itself $\to0$.

   Step-by-Step Solution

   1. For continuity at $x=0$: $k=\lim_{x\to0}\left(\frac{2+3x^2}{2+5x^2}\right)^{8/x^2+3}$.
   2. Let $L$ denote this limit. Then $\log L=\lim_{x\to0}\left(\frac{8}{x^2}+3\right)\log\left(\frac{2+3x^2}{2+5x^2}\right)$.
   3. Write the base as $1+u$ where $u=\frac{2+3x^2}{2+5x^2}-1=\frac{-2x^2}{2+5x^2}\approx -x^2$ for small $x$ (to leading order).
   4. $\log(1+u)\approx u\approx -x^2$ for small $x$. …

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4. AP EAPCET 2026Set eng-2026-05-15-FN1 markMCQ
   Q.If $f(x) = \begin{cases} \dfrac{x^2 - 4}{\sqrt{2 - x}}, & \text{for } x < 2 \\ a, & \text{for } x = 2 \\ \log(x - 2), & \text{for } x > 2 \end{cases}$ is a real valued function and 'a' is a finite real number, then (A) f is continuous at x = 2 when a = 0 (B) f is left continuous at x = 2 when a = 0 (C) f is right continuous at x = 2 when a = log 2 (D) f is not continuous at x = 1

   ›Reveal solutionSolution

   This tests one-sided limits at a piecewise junction; the left-hand limit at $x=2$ is $0$ while the right-hand limit diverges to $-\infty$, so option (B) is the only correct statement.

   Concept and Intuition

   For left continuity at $x=2$ we only need $\lim_{x\to2^-}f(x)=f(2)$; the behaviour on the other side is irrelevant. Since $\log(x-2)\to-\infty$ as $x\to2^+$, no finite value of $a$ can make $f$ right-continuous or fully continuous at $x=2$ — so those statements must be false.

   Step-by-Step Solution

   1. Left-hand limit: let $x=2-h,\ h\to0^+$. Then $x^2-4=(2-h)^2-4=h^2-4h=h(h-4)$ and $\sqrt{2-x}=\sqrt h$.
   2. So $\dfrac{x^2-4}{\sqrt{2-x}} = \dfrac{h(h-4)}{\sqrt h} = \sqrt h\,(h-4) \to 0\cdot(-4)=0$.
   3. Left continuity at $x=2$ requires $f(2)=a$ to equal this limit, i.e. $a=0$. So (B) is true.
   4. Right-hand limit: as $x\to2^+$, $x-2\to0^+$, so $\log(x-2)\to-\infty$ — unbounded, so no finite $a$ (in particular $a=\log2$) can match it. So (C) is false, and (A) (needing both sides equal) is also false. …

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5. AP EAPCET 2026Set eng-2026-05-15-FN1 markMCQ
   Q.If $f(x) = \dfrac{\cos^4 x - 1}{x^2}$, $x \ne 0$ and $f(0) = 2$ is a real valued function, then (A) $\displaystyle\lim_{x \to 0} f(x)$ does not exist (B) $\displaystyle\lim_{x \to 0} f(x) = 1$ (C) f is not continuous at x = 0 (D) f is continuous at x = 0

   ›Reveal solutionSolution

   This tests factoring $\cos^4x-1$ and using the standard limit $\frac{\sin x}{x}\to1$; the limit exists and equals $-2$, but $f(0)=2$, so $f$ fails continuity at $0$.

   Concept and Intuition

   $\cos^4x-1$ factors as a difference of squares twice: $\cos^4x-1=(\cos^2x-1)(\cos^2x+1)=-\sin^2x(\cos^2x+1)$. This lets us isolate the familiar $\left(\frac{\sin x}{x}\right)^2\to1$ building block.

   Step-by-Step Solution

   1. $\cos^4x - 1 = (\cos^2x-1)(\cos^2x+1) = -\sin^2x\,(\cos^2x+1)$.
   2. $f(x) = \dfrac{-\sin^2x(\cos^2x+1)}{x^2} = -\left(\dfrac{\sin x}{x}\right)^2(\cos^2x+1)$.
   3. As $x\to0$: $\left(\dfrac{\sin x}{x}\right)^2\to1$ and $\cos^2x+1\to2$.
   4. So $\displaystyle\lim_{x\to0}f(x) = -1\times2 = -2$ — this limit exists (rules out (A)) and is not $1$ (rules out (B)). …

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6. AP EAPCET 2026Set eng-2026-05-18-FN1 markMCQ
   Q.If $f(x)$ defined by $f(x) = \begin{cases} \dfrac{1 - \tan x}{4x - \pi}, & x \neq \dfrac{\pi}{4}, x \in \left[0, \dfrac{\pi}{2}\right] \\ k, & x = \dfrac{\pi}{4} \end{cases}$ is continuous in $\left[0, \dfrac{\pi}{2}\right]$, then $k =$ (A) $-1$ (B) $-\dfrac{1}{2}$ (C) $\dfrac{1}{2}$ (D) $1$

   ›Reveal solutionSolution

   Substituting $x=\pi/4+h$ and using the tangent subtraction identity, the limit of

   $\dfrac{1-\tan x}{4x-\pi}$ as $x\to\pi/4$ works out to $-\dfrac12$, which is the required

   value of $k$.

   Concept and Intuition

   For continuity at $x=\pi/4$, we need $k=\lim_{x\to\pi/4}f(x)$. Since the expression is $0/0$

   at $x=\pi/4$, shifting variables via $x=\pi/4+h$ (so $h\to0$) turns $\tan x$ into

   $\tan(\pi/4+h)$, which has a clean expansion using the tangent-addition formula.

   Step-by-Step Solution

   1. Let $x=\dfrac{\pi}{4}+h$, so $h\to0$ as $x\to\pi/4$.
   2. $\tan\left(\dfrac{\pi}{4}+h\right)=\dfrac{1+\tan h}{1-\tan h}$.
   3. $1-\tan x = 1-\frac{1+\tan h}{1-\tan h}=\frac{(1-\tan h)-(1+\tan h)}{1-\tan h}=\frac{-2\tan h}{1-\tan h}$
   4. $4x-\pi = 4\left(\dfrac{\pi}{4}+h\right)-\pi = 4h$.
   5. So

   $f(x)=\frac{-2\tan h}{4h(1-\tan h)}$

   6. As $h\to0$, $\tan h\sim h$, so …

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7. AP EAPCET 2025Set eng-2025-05-21-FN1 markMCQ
   Q.If a real valued function $f(x)=\begin{cases}\dfrac{x^2+(a+3)x+(a+1)}{x+3}, & x\neq -3\\ -\dfrac{5}{2}, & x=-3\end{cases}$ is continuous at $x=-3$, then $\displaystyle\lim_{x\to a}(x^2+x+1) =$ (A) $\dfrac{7}{4}$ (B) $\dfrac{5}{2}$ (C) $\dfrac{4}{7}$ (D) $\dfrac{2}{5}$

   ›Reveal solutionSolution

   This tests continuity of a rational function with a removable-type discontinuity: matching the limit to the given value pins down the parameter $a$, then the asked limit is a trivial polynomial evaluation. Answer: $\dfrac{7}{4}$.

   Concept and Intuition

   A piecewise function is continuous at a point if the limit of the "elsewhere" formula, as $x$ approaches that point, equals the value assigned at that point. Here the formula is a rational function whose denominator vanishes at $x=-3$; for the limit to exist (and be finite, matching $-\tfrac52$), the numerator must also vanish there, so that $(x+3)$ cancels — this is the standard "$\frac00$ removable singularity" idea.

   Step-by-Step Solution

   1. Continuity at $x=-3$ requires $\displaystyle\lim_{x\to -3}\frac{x^2+(a+3)x+(a+1)}{x+3} = -\frac52$.
   2. Since the denominator $\to 0$, the numerator must vanish at $x=-3$ (else the limit is $\pm\infty$, not finite):

   $(-3)^2+(a+3)(-3)+(a+1) = 9-3a-9+a+1 = 1-2a = 0 \implies a=\frac12.$

   3. Check: with $a=\tfrac12$, numerator $=x^2+\tfrac72x+\tfrac32$. Factor out $(x+3)$: $x^2+\tfrac72x+\tfrac32 = (x+3)\left(x+\tfrac12\right)$ (verify: $(x+3)(x+\tfrac12)=x^2+\tfrac12x+3x+\tfrac32=x^2+\tfrac72x+\tfrac32$ ✓).
   4. So $\displaystyle\lim_{x\to -3}\frac{(x+3)(x+\frac12)}{x+3} = -3+\frac12 = -\frac52$, matching the given value — confirming $a=\tfrac12$. …

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8. AP EAPCET 2025Set eng-2025-05-21-AN1 markMCQ
   Q.If a function f defined by $f(x) = \begin{cases} \dfrac{1 - \cos 4x}{x^2}, & x < 0 \\ a, & x = 0 \\ \dfrac{\sqrt{x}}{\sqrt{16 + \sqrt{x}} - 4}, & x > 0 \end{cases}$ is continuous at $x = 0$, then $a =$ (A) $8$ (B) $4$ (C) $3$ (D) $2$

   ›Reveal solutionSolution

   Both one-sided limits at $x=0$ evaluate to $8$ (via $1-\cos\theta=2\sin^2(\theta/2)$ on the left, and rationalizing on the right), so $a=8$.

   Concept and Intuition

   For $f$ to be continuous at $0$, the left-hand limit, right-hand limit, and $f(0)=a$ must all agree. The left piece is a classic $0/0$ trig limit solved via the half-angle identity for $1-\cos\theta$; the right piece is a classic surd limit solved by rationalizing the denominator.

   Step-by-Step Solution

   1. Left limit: $1-\cos4x=2\sin^2(2x)$, so $\dfrac{1-\cos4x}{x^2}=\dfrac{2\sin^2(2x)}{x^2}=8\left(\dfrac{\sin2x}{2x}\right)^2$.
   2. As $x\to0^-$, $\dfrac{\sin2x}{2x}\to1$, so this limit $\to8(1)^2=8$.
   3. Right limit: let $t=\sqrt x$ (so $t\to0^+$ as $x\to0^+$): $\dfrac{\sqrt x}{\sqrt{16+\sqrt x}-4}=\dfrac{t}{\sqrt{16+t}-4}$.
   4. Rationalize: multiply by $\dfrac{\sqrt{16+t}+4}{\sqrt{16+t}+4}$: $\dfrac{t(\sqrt{16+t}+4)}{(16+t)-16}=\dfrac{t(\sqrt{16+t}+4)}{t}=\sqrt{16+t}+4$. …

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9. AP EAPCET 2025Set eng-2025-05-23-AN1 markMCQ
   Q.If a real valued function $f(x) = \begin{cases} (1+\sin x)^{\operatorname{cosec}x}, & -\pi/2 < x < 0 \\ a, & x = 0 \\ \dfrac{e^{2/x}+e^{3/x}}{ae^{2/x}+be^{3/x}}, & 0 < x < \pi/2 \end{cases}$ is continuous at x = 0, then $ab =$ (A) $e$ (B) $e^2$ (C) 1 (D) $-1$

   ›Reveal solutionSolution

   The left-hand limit is the classical $1^\infty$ form giving $e$, fixing $a=e$;

   the right-hand limit needs dividing by the dominant exponential to fix $b$.

   Continuity forces both, and $ab=1$.

   Concept and Intuition

   A piecewise function is continuous at a point only if the left-hand limit,

   right-hand limit, and the function's value there all agree. Here the left

   piece is a $1^\infty$ indeterminate form (standard trick: exponentiate and use

   $\log(1+u)\sim u$), and the right piece is a ratio of two exponentials growing at

   different rates as $x\to0^+$ (since $1/x\to+\infty$), so the faster-growing

   exponential $e^{3/x}$ dominates and everything else becomes negligible after

   dividing through by it.

   Step-by-Step Solution

   1. Left limit: $L=\lim_{x\to0^-}(1+\sin x)^{\operatorname{cosec}x}$. Take logs: $\log L=\lim \operatorname{cosec}x\cdot\log(1+\sin x)=\lim\dfrac{\sin x}{\sin x}\left(1+O(\sin x)\right)\to1$. So $L=e$. Continuity requires $f(0)=a=L=e$.
   2. Right limit: $R=\lim_{x\to0^+}\dfrac{e^{2/x}+e^{3/x}}{ae^{2/x}+be^{3/x}}$. Divide numerator and denominator by $e^{3/x}$: $R=\lim_{x\to0^+}\dfrac{e^{-1/x}+1}{ae^{-1/x}+b}$. …

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10. AP EAPCET 2025Set eng-2025-05-24-FN1 markMCQ
    Q.If $f(x) = \begin{cases} \dfrac{(e^{ax} - 1)\log(1+x)}{\sin^2 x}, & \text{if } x > 0 \\ 2, & \text{if } x = 0 \\ \dfrac{\cos 4x - \cos bx}{\tan^2 x}, & \text{if } x < 0 \end{cases}$ is continuous at $x = 0$ then $\sqrt{b^2 - a^2} =$ (A) $4$ (B) $5$ (C) $3$ (D) $7$

    ›Reveal solutionSolution

    Continuity at $x=0$ forces both one-sided limits to equal $f(0)=2$; solving gives $a=2$, $b^2=20$, so $\sqrt{b^2-a^2}=4$.

    Concept and Intuition

    For $f$ to be continuous at $0$, we need $\displaystyle\lim_{x\to0^+}f(x)=\lim_{x\to0^-}f(x)=f(0)=2$. Each one-sided piece is a $0/0$ form that resolves using the standard small-angle equivalences $e^{t}-1\sim t$, $\log(1+t)\sim t$, $\sin t\sim t$, $\tan t \sim t$, and $\cos t\approx 1-\dfrac{t^2}{2}$.

    Step-by-Step Solution

    1. Right-hand limit ($x\to0^+$): using $e^{ax}-1\sim ax$, $\log(1+x)\sim x$, $\sin^2x\sim x^2$,

    $\lim_{x\to0^+}\frac{(e^{ax}-1)\log(1+x)}{\sin^2x}=\lim_{x\to0^+}\frac{(ax)(x)}{x^2}=a.$

    Setting this equal to $f(0)=2$: $a=2$.

    2. Left-hand limit ($x\to0^-$): expand $\cos4x\approx 1-\dfrac{(4x)^2}{2}=1-8x^2$ and $\cos(bx)\approx1-\dfrac{b^2x^2}{2}$, and $\tan^2x\sim x^2$:

    $\cos4x-\cos(bx)\approx \left(1-8x^2\right)-\left(1-\frac{b^2x^2}{2}\right)=x^2\left(\frac{b^2}{2}-8\right).$ …

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11. AP EAPCET 2025Set eng-2025-05-27-FN1 markMCQ
    Q.If the function f defined by $f(x) = \begin{cases} \dfrac{1 - \cos 4x}{x^2}, & x < 0 \\ a, & x = 0 \\ \dfrac{\sqrt{x}}{\sqrt{16 + \sqrt{x}} - 4}, & x > 0 \end{cases}$ is continuous at $x = 0$, then $a =$ (A) $1$ (B) $2$ (C) $4$ (D) $8$

    ›Reveal solutionSolution

    Continuity at $x=0$ requires the left limit, the right limit, and $f(0)=a$ to all agree; both one-sided limits work out to $8$. Answer: $a=8$.

    Concept and Intuition

    For a piecewise function to be continuous at a junction point, the value defined there ($a$) must equal both the limit approaching from the left and the limit approaching from the right. Each side here needs a standard trick: the trig side uses the double-angle identity $1-\cos\theta=2\sin^2(\theta/2)$, and the surd side needs rationalisation to remove the $\sqrt{\ }-\sqrt{\ }$ indeterminate form.

    Step-by-Step Solution

    1. Left-hand limit ($x\to0^-$): $1-\cos4x=2\sin^2(2x)$, so

    $\lim_{x\to0^-}\frac{1-\cos4x}{x^2}=\lim_{x\to0^-}\frac{2\sin^2(2x)}{x^2}=2\lim_{x\to0}\left(\frac{\sin2x}{2x}\right)^2\cdot4=2\cdot1\cdot4=8.$

    2. Right-hand limit ($x\to0^+$): rationalise $\dfrac{\sqrt x}{\sqrt{16+\sqrt x}-4}$ by multiplying top and bottom by $\sqrt{16+\sqrt x}+4$: $\frac{\sqrt x\,(\sqrt{16+\sqrt x}+4)}{(16+\sqrt x)-16}=\frac{\sqrt x\,(\sqrt{16+\sqrt x}+4)}{\sqrt x}=\sqrt{16+\sqrt x}+4.$ …

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12. AP EAPCET 2024Set eng-2024-05-18-FN1 markMCQ
    Q.$f(x)=\begin{cases}\dfrac{(2x^2-ax+1)-(ax^2+3bx+2)}{x+1}, & \text{if } x\neq -1\\ k, & \text{if } x=-1\end{cases}$ is a real valued function. If $a,b,k\in R$ and $f$ is continuous on $R$ then $k=$ (A) $-\dfrac{1}{3}$ (B) 6 (C) $a-2$ (D) $a-3$

    ›Reveal solutionSolution

    Continuity at the removable-type point requires the numerator to vanish there (fixing $b$) and then simplifying the quotient to evaluate $k$ as a limit. The answer is $k=a-3$.

    Concept and Intuition

    A rational expression $\dfrac{N(x)}{x+1}$ can only have a finite limit as $x\to-1$ if $N(-1)=0$ (otherwise the limit is $\pm\infty$ and no choice of $k$ can make $f$ continuous there). So the first job is to use that vanishing condition to pin down any free constant, then simplify by cancelling the common factor.

    Step-by-Step Solution

    1. Numerator: $(2x^2-ax+1)-(ax^2+3bx+2)=(2-a)x^2-(a+3b)x-1$.
    2. At $x=-1$: $(2-a)(1)+(a+3b)-1=1+3b$. For the limit (hence continuity) to exist finitely, this must be $0$: $b=-\dfrac13$.
    3. With $b=-1/3$, numerator becomes $(2-a)x^2-(a-1)x-1$.
    4. Factor out $(x+1)$: writing $(2-a)x^2-(a-1)x-1=(x+1)\left[(2-a)x-1\right]$ (verified by expansion, matching all coefficients). …

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