TS EAMCET Limits and Continuity — practice questions
106 free MCQs with worked solutions. Tap any question for the answer + explanation, or practice them all in the app.
Practice TS EAMCET Limits and Continuity in the app →Evaluate $\displaystyle\lim_{x \to 0} \dfrac{ in 3x}{x}$.Evaluate $\displaystyle\lim_{x \to 2} \dfrac{x^2 - 4}{x - 2}$.Evaluate $\displaystyle\lim_{x \to 0} \dfrac{1 - \cos 2x}{x^2}$.Evaluate $\displaystyle\lim_{x \to \infty} \left(1 + \dfrac{2}{x}\right)^x$.Consider $f(x) = |x|$ at $x = 0$. Which statement is correct?$ in(180° - \theta)$ equals:$ in^2\theta + \cos^2\theta$ equals:$ in(A + B) = $A **random variable** is best defined as:A **discrete** random variable is one whose possible values:For a continuous random variable $X$, the probability that $X$ equals any specific value is:The **cumulative distribution function (c.d.f.)** $F(x)$ of a discrete random variable is:When two fair dice are rolled and $X$ denotes the sum of the upper faces, the range of $X$ is:If a fair coin is tossed twice and $X$ = number of heads, then $P[X = 1]$ equals:If two cards are drawn with replacement from a deck of 52, $P[X = 2]$ where $X$ = number of aces equals:A fair die is thrown. Let $X$ = number of factors of the number on the upper face. $P[X = 2]$ equals:$f(x) = 3x^2$ for $0 < x < 1$ is the p.d.f. of $X$. The probability $P[1/2 < X < 1]$ is:If $f(x) = kx^2(1 - x)$ for $0 < x < 1$ is the p.d.f. of $X$, the value of $k$ is:For p.d.f. $f(x) = x/8$, $0 < x < 4$, the probability $P[x < 1.5]$ is:Find $k$ if $f(x) = kx$ for $0 < x < 2$ and 0 otherwise is a valid p.d.f.The c.d.f. $F(x)$ of the p.d.f. $f(x) = 3x^2$ for $0 < x < 1$ is:Two cards are drawn with replacement. $X$ = number of aces. The probability $P[X \geq 1]$ equals:For p.d.f. $f(x) = x^2/3$ for $-1 < x < 2$, the probability $P[0 < X \leq 1]$ equals:For p.d.f. $f(x) = k(4 - x^2)$ for $-2 \leq x \leq 2$ (and 0 elsewhere), the value of $k$ is:For the c.d.f. $F(x) = 3x - 2x^3$ (from p.d.f. $f(x) = 3(1 - 2x^2)$ on $[0, 1]$), $F(0.5)$ equals:If a fair coin is tossed $n$ times, the probability of any exact pattern (like all heads) is:If a die is thrown and 'getting an odd number' is success, then $p$ and $q$ are:A bag has balls marked 0–9. Four balls drawn with replacement. Probability that **none** is marked '0' is:Probability of bomb hitting target = 0.8. Out of 10 bombs dropped, probability that **exactly 2 miss** equals sin(0°) equals:cos(90°) equals:sin²θ + cos²θ equals:tan(45°) equals:In which quadrant is sin θ positive and cos θ negative?lim_(x→2) (x² - 4)/(x - 2) =lim_(x→0) sin(x)/x =For limit to EXIST at x = a, left-hand and right-hand limits must:A function f is continuous at x = a if:d/dx (xⁿ) = ?lim_(x→0) (1 - cos x)/x² =lim_(x→∞) (1 + 1/x)^x =lim_(x→0) (e^x - 1)/x =lim_(x→0) (sin 3x)/(sin 5x) =d/dx (sin x) =d/dx (eˣ) =d/dx (ln x) =Find d/dx (x² sin x) by product rule:By chain rule, d/dx [sin(x²)] =lim_(x→∞) (3x² - 5)/(7x² + 2) =l'Hôpital's rule applies to limits of form:Find dy/dx by implicit differentiation for x² + y² = 25:d/dx (xˣ) where x > 0:For f(x) = sin(1/x) for x ≠ 0, what is the limit as x → 0?Differentiation by chain rule for d/dx [ln(sin x)]:lim_(x→0) [(1 + x)^(1/x)] =Higher-order derivative: d²/dx² (x⁴) =f(x) is discontinuous at x = a if which condition fails?For y = x³ - 6x² + 9x, find critical points (where y' = 0):By Taylor expansion around x = 0: e^x = 1 + x + ___:As x creeps toward 2, which number does (x² − 4)/(x − 2) settle at?The standard result for lim(x→0) sin x / x, with x in radians, is which value?Near zero, the ratio (1 − cos x)/x² gets close to what number?lim(x→0) tan x / x is equal to which of these?Compute lim(x→0) sin 3x / x.When x gets very small, sin 5x divided by sin 2x approaches which fraction?Which number is the limit of ln(1 + x)/x when x shrinks to zero?Find lim(x→0) of (3ˣ − 1)/x.Which constant is defined by the behaviour of (1 + 1/x)ˣ for huge x?The product (1 + x) raised to the power 1/x approaches what as x nears 0?If x → ∞, (1 + 2/x)ˣ converges to which value?Evaluate: lim(x→0) (1 + 3x)^(1/x).Rationalise to find lim(x→0) [√(1 + x) − 1]/x.Using the xⁿ − aⁿ formula, (x⁵ − 32)/(x − 2) has which limit as x → 2?Squeezed between −|x| and |x|, the product x sin(1/x) tends to what at 0?As x runs off to infinity, sin x divided by x tends to which value?Approaching 0 through positive values only, the reciprocal 1/x heads toward what?For the greatest integer function ⌊x⌋, what value do inputs just below 2 give?Approaching 2 from the right, the floor function ⌊x⌋ takes which constant value?Using small-angle behaviour, (tan 2x − sin 2x)/x³ approaches what at x = 0?Write (e³ˣ − 1)/x as 3 times a standard form; its limit at 0 is then what?Which number does (1 − cos 4x)/x² approach as x shrinks to zero?Taylor's series for sin x shows (sin x − x)/x³ tends to what at 0?After expanding eˣ as a power series, (eˣ − 1 − x)/x² has which limit at 0?Recognising a derivative, ln x/(x − 1) tends to what as x nears 1?By rationalising, what is the large-x limit of √(x² + x) − x?The gap between √(x + 1) and √x shrinks to what value as x becomes huge?The sum of the first n natural numbers, divided by n², tends to what as n grows?Sum the squares 1² to n² and divide by n³. What fraction appears in the limit?Which of these is an indeterminate form?With L'Hospital's rule, what does x ln x tend to as x → 0 from the right?Seen as a derivative at π/4, (sin x − cos x)/(x − π/4) tends to what?Factor and cancel x − 1 in (x² − 1)/(x² − 3x + 2). What limit results at x = 1?Cancelling x − 2 in (x³ − 8)/(x² − 4) gives what limit at x = 2?After rationalising the numerator, [√(1 + x) − √(1 − x)]/x tends to what at 0?The product x cot x settles at which value as x moves to 0?For f(x) = sin 2x / x with x ≠ 0, which value of f(0) makes f continuous?A piecewise rule uses kx + 1 up to x = 5 and 3x − 5 beyond it. For continuity at 5, what is k?Take f = kx² when x ≤ 2 and f = 3 when x > 2. Continuity at 2 forces k to be what?Define f(x) = (1 − cos x)/x² off zero. Continuity at 0 needs f(0) equal to what?At which real number does f(x) = 1/(x − 3) break down?At which of these points is tan x discontinuous?How many points in the open interval (0, 3) is ⌊x⌋ discontinuous at?For f(x) = x³ + x − 1, in which interval must a root lie?What does the differentiability of a function at a point guarantee about it there?Let f(x) = 2x + 1 for x < 2 and x² + 1 for x ≥ 2. What is the limit at x = 2?How many real points of discontinuity does (x + 1)/(x² − 5x + 6) have?