Chapter 14: Limits, Continuity and Differentiability (Set-2)
Which statement defines limx→af(x)=Llimx→af(x)=L
A f(x) near L
B f(a) equals L
C x equals a
D f(x) constant
The limit means that for x sufficiently close to a (not necessarily equal), f(x) becomes as close as desired to L. It describes behavior near a.
If limx→a−f(x)=2limx→a−f(x)=2 and limx→a+f(x)=2limx→a+f(x)=2, then
A Limit equals 0
B Limit infinite
C Limit equals 2
D Limit absent
When both one-sided limits exist and are equal to the same number, the two-sided limit exists and equals that common value, here 2.
A limit can exist even if f(a) is
A Defined equal
B Not defined
C Always zero
D Always one
The limit depends on values of f(x) near a, not necessarily at a. A “hole” at x=a can still have a well-defined limit.
Which is a limit law
A Limit of colour
B Limit of length
C Limit of sum
D Limit of angle
Limit laws include sum, difference, product, and quotient (when denominator limit is nonzero). They allow computing limits by splitting expressions into simpler parts.
If limf(x)=3limf(x)=3 and limg(x)=4limg(x)=4, then lim(f+g)lim(f+g) equals
A 12
B 7
C 1
D 0
By the sum law, lim(f(x)+g(x))=limf(x)+limg(x)lim(f(x)+g(x))=limf(x)+limg(x), provided both limits exist. So the result is 3+4=7.
If limf(x)=5limf(x)=5, then lim(2f(x))lim(2f(x)) equals
A 10
B 7
C 3
D 5
By constant multiple law, lim(cf(x))=climf(x)lim(cf(x))=climf(x). With c=2 and limit 5, the new limit is 10.
If limf(x)=2limf(x)=2 and limg(x)=3limg(x)=3, then lim(f⋅g)lim(f⋅g) equals
A 5
B 1
C 0
D 6
Product law states lim(fg)=(limf)(limg)lim(fg)=(limf)(limg) when both limits exist. Here 2×3 gives 6.
If limg(x)≠0limg(x)=0, then limfglimgf equals
A limf−limglimf−limg
B limf+limglimf+limg
C limflimglimglimf
D Always zero
Quotient law holds when the denominator limit is nonzero. Then the limit of a quotient equals the quotient of the limits.
Standard limit limx→0tanxxlimx→0xtanx equals
A 0
B 1
C 2
D −1
Using tanx=sinxcosxtanx=cosxsinx and standard limits, tanxx=sinxx⋅1cosx→1⋅1=1xtanx=xsinx⋅cosx1→1⋅1=1.
limx→0ex−1xlimx→0xex−1 equals
A 0
B e
C 1
D 2
A key standard limit is limx→0ex−1x=1limx→0xex−1=1. It follows from the series expansion ex=1+x+…ex=1+x+….
limx→0ln(1+x)xlimx→0xln(1+x) equals
A 1
B 0
C 2
D −1
Near 0, ln(1+x)=x−x22+…ln(1+x)=x−2×2+…. Dividing by x gives 1−x2+…1−2x+…, so the limit is 1.
If f is continuous at x=a, then limx→af(x)limx→af(x) equals
A 0
B f(a)
C ∞
D Not exist
Continuity at a means the limit at a exists and equals the function value. So limx→af(x)=f(a)limx→af(x)=f(a).
A function continuous on [a,b] is continuous at
A Only a and b
B Only middle
C No point
D Every point
Continuity on a closed interval means continuity at each interior point and one-sided continuity at endpoints. Overall, it is continuous everywhere on [a,b].
Which function is continuous everywhere
A Step function
B 1/x
C Polynomial function
D tan x
Polynomials are continuous for all real x. The others have discontinuities: step functions jump, 1/x breaks at 0, and tan x breaks at odd π/2π/2.
Rational function is continuous where
A Numerator nonzero
B Denominator nonzero
C Derivative exists
D Always continuous
A rational function p(x)q(x)q(x)p(x) is continuous at all points where it is defined. It is undefined when q(x)=0, causing discontinuity there.
If LHL exists but RHL fails, then two-sided limit
A Exists always
B Equals LHL
C Does not exist
D Equals zero
A two-sided limit requires both one-sided limits to exist and be equal. If either side fails, the overall limit does not exist.
Which discontinuity may be fixed by redefining value
A Removable
B Jump
C Infinite
D Oscillatory
In removable discontinuity, the limit exists but the function value is missing or different. Setting f(a) equal to the limit makes the function continuous.
Discontinuity of x2−1x−1x−1×2−1 at x=1 is
A Jump
B Removable
C Infinite
D Oscillatory
x2−1x−1=(x−1)(x+1)x−1=x+1x−1×2−1=x−1(x−1)(x+1)=x+1 for x≠1. The limit at 1 is 2, but original is undefined at 1, so removable.
The function 1xx1 has at x=0
A Removable break
B Jump break
C Infinite break
D No break
As x→0, 1/x becomes unbounded. It approaches ±∞ depending on side, indicating a vertical asymptote and an infinite discontinuity.
A jump discontinuity shows in graph as
A Smooth curve
B Hole only
C Tangent flat
D Sudden step
Jump discontinuity appears as a break where the function “jumps” from one value to another. One-sided limits exist but are unequal.
Differentiability at x=a means
A f(a) exists
B Derivative exists
C Limit infinite
D Value zero
A function is differentiable at a if the derivative f′(a)f′(a) exists. This requires the difference quotient to approach a single finite value.
Differentiability implies continuity because
A Continuity implies slope
B Graph always straight
C Derivative uses limit
D f(a) must be zero
The derivative is defined using a limit of f(a+h)−f(a)hhf(a+h)−f(a). If this limit exists, it forces f(x)→f(a) as h→0, ensuring continuity.
A common non-differentiable point is
A Corner
B Smooth point
C Polynomial root
D Constant region
At a corner, left and right derivatives are different, so the slope is not unique. The function may still be continuous, but derivative does not exist.
Function f(x)=∣x−2∣f(x)=∣x−2∣ is non-differentiable at
A x=0
B x=2
C x=1
D x=3
|x−2| has a sharp corner at x=2. Left slope is −1 and right slope is +1, so derivative fails at x=2 though the function remains continuous.
Derivative measures
A Average value
B Total area
C Total length
D Instant rate
The derivative gives instantaneous rate of change of the function with respect to x. Geometrically, it is slope of the tangent at a point.
Which is first principle formula
A limf(x)xlimxf(x)
B limf(x+h)limf(x+h)
C limf(x+h)−f(x)hlimhf(x+h)−f(x)
D limhf(x)limhf(x)
The derivative from first principles is defined by the limit of the difference quotient as h approaches 0. It captures tangent slope without using rules.
Second derivative is written as
A f′(x)
B f″(x)
C f(x)
D f⁻¹(x)
The second derivative is the derivative of f′(x) and is denoted f″(x). It indicates concavity and how the slope changes with x.
If f″(x) > 0 on interval, curve is
A Concave down
B Constant
C Concave up
D Discontinuous
A positive second derivative means the slope is increasing, so the graph bends upward. This is called concave up (or convex).
If f″(x) < 0 on interval, curve is
A Concave down
B Concave up
C Constant
D Vertical line
A negative second derivative means the slope decreases as x increases. The curve bends downward, which is concave down.
Differentiation of parametric form gives
A dy/dx = dx/dt ÷ dy/dt
B dy/dx = dy/dt ÷ dx/dt
C dy/dx = dx·dy
D dy/dx = x+y
For x=x(t) and y=y(t), the derivative is dydx=dy/dtdx/dtdxdy=dx/dtdy/dt, assuming dx/dt ≠ 0. This is basic parametric differentiation.
If y is given by x²+y²=1, then use
A Product rule
B Only quotient rule
C Implicit method
D Only limits
Here y is not isolated as a function of x. We differentiate both sides with respect to x, treating y as y(x), and use dy/dx where needed.
A simple use of logarithmic differentiation is for
A x+1 type
B x−2 type
C 2x+3 type
D x^x type
For y=x^x, taking ln gives ln y = x ln x, which is easier to differentiate. This method handles variable bases and variable exponents smoothly.
Indeterminate form for 0000 means
A Limit fixed
B Form unclear
C Limit infinite
D Limit negative
0/0 does not tell the limit. The expression may simplify to many possible limits. We must factor, rationalize, or apply standard methods to find the actual value.
Indeterminate form for ∞∞∞∞ means
A Form unclear
B Always one
C Always infinity
D Always zero
∞/∞ is not definite because the numerator and denominator may grow at different rates. We simplify or use L’Hospital’s rule to compare growth.
Converting 0·∞ form is done by
A Multiply more
B Replace with 0
C Rewrite as quotient
D Replace with ∞
We rewrite a product into a fraction, like 0⋅∞0⋅∞ into 0000 or ∞∞∞∞ forms using algebra, making limit techniques applicable.
Rationalization helps mainly in limits with
A Polynomials only
B Square roots
C Constants only
D Trig only
When limits involve expressions like x−ax−a, multiplying by the conjugate removes radicals in numerator or denominator and often simplifies 0/0 forms.
For large x, limit of 3×2+1×2−5×2−53×2+1 is
A 0
B ∞
C 3
D 1/3
Divide numerator and denominator by x²: 3+1/x21−5/x2→3+01−0=31−5/x23+1/x2→1−03+0=3. This is a basic limit at infinity.
Comparison of growth: as x→∞, which grows faster
A x
B log x
C constant
D 1/x
As x becomes large, x increases much faster than log x, while constants stay fixed and 1/x goes to 0. Growth order is constant < log x < x.
Continuity of sinxsinx holds for
A Only x>0
B All real x
C Only integers
D Only near 0
Trigonometric functions like sin x and cos x are continuous for all real values. There are no breaks or holes in their graphs.
Discontinuity of tanxtanx happens at
A multiples of ππ
B only at 0
C odd π/2π/2
D only at 1
tanx=sinxcosxtanx=cosxsinx is undefined where cosx=0cosx=0, i.e., at x=π2+kπx=2π+kπ. These points create infinite discontinuities.
A function continuous on (a,b) means
A continuous at endpoints
B differentiable always
C limit always zero
D continuous at each point
Continuity on an open interval requires the function to be continuous at every point inside it. Endpoint checks are not required because endpoints are not included.
Piecewise function can be continuous if
A parts always equal
B limits match value
C derivative always zero
D only one side exists
At the joining point, we must ensure LHL = RHL and equal to the defined value. If these conditions hold, the piecewise function becomes continuous there.
For differentiability of a piecewise function at join, need
A left slope equals right slope
B only continuity
C f(a) undefined
D jump in graph
Differentiability at the joining point requires continuity plus equality of left and right derivatives. Even if continuous, unequal slopes cause a corner and derivative fails.
If f(x) is differentiable on interval, then f(x) is
A discontinuous there
B oscillatory there
C continuous there
D undefined there
Differentiability implies continuity at every point. If a function is differentiable throughout an interval, it must also be continuous throughout that interval.
Which describes “limit from left” at a
A x→a⁺
B x→a⁻
C x=a
D x→∞
Left-hand limit considers x approaching a from values less than a, denoted x→a⁻. It captures behavior just to the left of the point.
Which describes “limit from right” at a
A x→a⁻
B x=a
C x→−∞
D x→a⁺
Right-hand limit considers x approaching a from values greater than a, denoted x→a⁺. It checks behavior immediately to the right side.
If a function is increasing and differentiable, then f′(x) is generally
A Always negative
B Always positive
C Nonnegative
D Always undefined
For an increasing differentiable function, the slope cannot be negative. It can be positive or sometimes zero at flat points, so the correct condition is f′(x) ≥ 0.
Intermediate Value Theorem needs function to be
A differentiable
B continuous
C periodic
D constant
IVT applies to continuous functions on an interval. It guarantees that the function takes every value between f(a) and f(b), ensuring no “gaps” in outputs.
Rolle’s theorem requires
A both A and B
B f(a)=f(b)
C continuity and differentiability
D only derivative
Rolle’s theorem needs f continuous on [a,b], differentiable on (a,b), and f(a)=f(b). Then at some point c, f′(c)=0.
A basic limit of sequence idea is
A n→0 behavior
B x→a behavior
C n→∞ behavior
D y→a behavior
For sequences, we study terms as n becomes very large. If the terms approach a fixed number, the sequence has a limit, similar to function limits at infinity.