Q.Find whether the function is continuous or discontinuous at the indicated point: $f(x) = \begin{cases} \dfrac{|x - 4|}{2(x - 4)}, & x \ne 4 \\ 0, & x = 4 \end{cases}$ at $x = 4$.

Showing the 12 most recent of 32 on this concept.

1. 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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2. 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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3. 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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4. 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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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 2024Set eng-2024-05-22-FN1 markMCQ
   Q.Let $f(x) = \begin{cases} 0, & x = 0 \\ 2 - x, & \text{for } 0 < x < 1 \\ 2, & \text{for } x = 1 \\ \dfrac{1}{2} - x, & \text{for } 1 < x < 2 \\ \dfrac{-3}{2}, & \text{for } x \ge 2 \end{cases}$ then which of the following is true (A) $f$ is right continuous at $x = 0$ (B) $f$ is left continuous at $x = 1$ (C) $f$ is right continuous at $x = 1$ (D) $f$ is continuous at $x = 2$

   ›Reveal solutionSolution

   Compute the left-hand limit, right-hand limit, and function value at each candidate point and compare; only at $x=2$ do all three coincide.

   Concept and Intuition

   A function is continuous at a point if the left-hand limit, right-hand limit, and the function's value there all agree. "Right continuous" only needs the right-hand limit to equal the function value; "left continuous" only needs the left-hand limit to match.

   Step-by-Step Solution

   1. At $x=0$: $f(0)=0$. Right-hand limit: $\lim_{x\to0^+}(2-x)=2$. Since $2\ne0$, $f$ is not right continuous at $0$ — (A) is false.
   2. At $x=1$: $f(1)=2$. Left-hand limit: $\lim_{x\to1^-}(2-x)=1$. Since $1\ne2$, not left continuous — (B) is false. Right-hand limit: $\lim_{x\to1^+}\left(\dfrac12-x\right)=-\dfrac12$. Since $-\dfrac12\ne2$, not right continuous either — (C) is false. …

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7. AP EAPCET 2023Set eng-2023-05-15-FN1 markMCQ
   Q.If $f:\mathbb{R}\to\mathbb{R}$ defined by $f(x)=\begin{cases} \dfrac{\sin x - \sin\dfrac{x}{2}}{x}, & x<0 \\[2mm] \dfrac{\sqrt{x^2+x}-\sqrt{x}}{x^{3/2}}, & x>0 \end{cases}$ is continuous on $\mathbb{R}$, then $f(0)=$ (A) $1/2$ (B) $3/2$ (C) $1$ (D) $-1$

   ›Reveal solutionSolution

   This tests continuity of a piecewise function at the junction point using standard small-$x$ approximations; both one-sided limits equal $1/2$.

   Concept and Intuition

   A piecewise function is continuous at a boundary point only if the left-hand limit, right-hand limit, and the defined value at that point all agree. Here $f(0)$ isn't given directly by either branch (both blow up as $0/0$), so we must compute the limits from each side and set $f(0)$ equal to their common value.

   Step-by-Step Solution

   1. Left-hand limit ($x\to 0^-$):

   $\lim_{x\to0^-}\frac{\sin x-\sin(x/2)}{x}$

   Use the small-angle expansion $\sin t = t - \dfrac{t^3}{6}+\cdots$:

   $\sin x - \sin\frac{x}{2} = \left(x-\frac{x^3}{6}\right)-\left(\frac{x}{2}-\frac{x^3}{48}\right)+\cdots = \frac{x}{2} + O(x^3)$

   Dividing by $x$: the limit is $\dfrac12$.

   2. Right-hand limit ($x\to 0^+$):

   $\lim_{x\to0^+}\frac{\sqrt{x^2+x}-\sqrt{x}}{x^{3/2}} = \lim_{x\to0^+}\frac{\sqrt{x}\left(\sqrt{x+1}-1\right)}{x^{3/2}} = \lim_{x\to0^+}\frac{\sqrt{x+1}-1}{x}$

   Using $\sqrt{1+x}\approx 1+\dfrac{x}{2}-\dfrac{x^2}{8}+\cdots$:

   $\frac{\sqrt{x+1}-1}{x}\approx \frac{x/2}{x} = \frac12$ …

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8. AP EAPCET 2022Set eng-2022-07-06-FN1 markMCQ
   Q.If $f(x) = \dfrac{(1-\sin x)}{\log(1+\pi^2 - 4\pi x + 4x^2)}$ is continuous at $x = \pi/2$, then $f(\pi/2) =$ (A) $\dfrac{1}{4}$ (B) $\dfrac{1}{8}$ (C) $\dfrac{1}{16}$ (D) $\dfrac{1}{32}$

   ›Reveal solutionSolution

   Recognising $1+\pi^2-4\pi x+4x^2$ as $1+(2x-\pi)^2$ turns this into a small-angle limit; the continuity value is $1/8$.

   Concept and Intuition

   For $f$ to be continuous at $x=\pi/2$, $f(\pi/2)$ must equal $\lim_{x\to\pi/2} f(x)$. The denominator's quadratic in $x$ is a perfect "sum-of-squares" shift once you notice $\pi^2-4\pi x+4x^2=(2x-\pi)^2$, turning this into a standard small-$t$ limit using $1-\cos t\approx t^2/2$ and $\log(1+u)\approx u$.

   Step-by-Step Solution

   1. Rewrite the denominator: $1+\pi^2-4\pi x+4x^2 = 1+(2x-\pi)^2$.
   2. Let $t=x-\pi/2$, so $x\to\pi/2 \iff t\to 0$, and $2x-\pi=2t$.
   3. Numerator: $1-\sin x = 1-\sin(\pi/2+t) = 1-\cos t$. For small $t$, $1-\cos t \approx \dfrac{t^2}{2}$.
   4. Denominator: $\log(1+(2t)^2) = \log(1+4t^2) \approx 4t^2$ for small $t$ (since $\log(1+u)\approx u$). …

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9. 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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10. AP EAPCET 2024Set eng-2024-05-19-AN1 markMCQ
    Q.The function $f(x)=\begin{cases}\dfrac{2}{5-x}, & x<3\\ 5-x, & x \ge 3\end{cases}$ is (A) left discontinuous at $x=3$ (B) left continuous at $x=3$ (C) right discontinuous at $x=5$ (D) discontinuous at $x=5$

    ›Reveal solutionSolution

    Checking the one-sided limits at the junction $x=3$ shows the function jumps from $1$ to $2$ on approach from the left, i.e. it is left-discontinuous (but right-continuous) at $x=3$; there is nothing special happening at $x=5$.

    Concept and Intuition

    A piecewise function can fail to be continuous from only one side at a junction point if the junction point itself is included in one branch. Here $x=3$ is the boundary, and it belongs to the $5-x$ branch (since that branch is defined for $x\ge3$), so we must check whether the other branch's limit (from the left) agrees with the actual function value.

    Step-by-Step Solution

    1. Function value at $x=3$: since $x\ge3$ uses $f(x)=5-x$, $f(3)=5-3=2$.
    2. Left-hand limit as $x\to3^-$: uses $f(x)=\dfrac{2}{5-x}$, so $\lim_{x\to3^-}\dfrac{2}{5-x}=\dfrac{2}{5-3}=1$.
    3. Since LHL $=1\ne f(3)=2$, the function is discontinuous when approached from the left — i.e. left discontinuous at $x=3$.
    4. Right-hand limit as $x\to3^+$: still $5-x\to2=f(3)$, so it is right-continuous at $x=3$ — this rules out "left continuous" (option B, false). …

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11. AP EAPCET 2023Set eng-2023-05-18-FN1 markMCQ
    Q.Which one of the following function is discontinuous at x=1? (A) $f(x)=\sin^2x+\tan^2x+\cos^2x-\sec^2x$ (B) $f(x)=\dfrac{1}{1+2^{\sin x}}$ (C) $f(x)=\begin{cases}\dfrac{x-1}{|x-1|+2(x-1)^2}, & x\neq1\\ 1, & x=1\end{cases}$ (D) $f(x)=e^{x}+5$

    ›Reveal solutionSolution

    Options (A), (B), (D) simplify to functions that are continuous everywhere; option (C)'s one-sided limits at $x=1$ disagree ($+1$ from the right, $-1$ from the left), so it alone is discontinuous at $x=1$.

    Concept and Intuition

    For piecewise/rational-looking expressions, check whether the identity $\sin^2x+\cos^2x=1$ and $\sec^2x-\tan^2x=1$ collapse them to constants (continuous), whether a denominator can vanish, and — for absolute-value expressions — whether the left and right limits genuinely agree.

    Step-by-Step Solution

    1. (A) $f(x)=\sin^2x+\tan^2x+\cos^2x-\sec^2x=(\sin^2x+\cos^2x)-(\sec^2x-\tan^2x)=1-1=0$ for all $x$ where $\tan,\sec$ are defined; at $x=1$ (radian) $\cos1\ne0$, so it's fine and continuous ($\equiv0$ near $x=1$).
    2. (B) $f(x)=\dfrac{1}{1+2^{\sin x}}$: since $2^{\sin x}>0$ always, the denominator never vanishes ($\ge1+2^{-1}=1.5$); continuous everywhere, including $x=1$.
    3. (C) For $x\ne1$, let $t=x-1$: $f=\dfrac{t}{|t|+2t^2}$.
       • As $t\to0^+$ ($x\to1^+$): $|t|=t$, so $f=\dfrac{t}{t+2t^2}=\dfrac{1}{1+2t}\to1$.
       • As $t\to0^-$ ($x\to1^-$): $|t|=-t$, so $f=\dfrac{t}{-t+2t^2}=\dfrac{1}{-1+2t}\to-1$. …

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12. 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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