Chapter 14: Limits, Continuity and Differentiability (Set-1)
If LHL = RHL = 5 at x=a, then limit at a equals
A 5
B Does not exist
C 0
D Infinity
A limit exists at x=a when the left-hand and right-hand limits are equal. If both equal 5, then the two-sided limit is 5.
Which must be checked for limx→af(x)limx→af(x) to exist
A Only f(a)
B Only LHL
C Only RHL
D LHL and RHL
The two-sided limit exists only when both one-sided limits exist and are equal. The actual value f(a) is not required for limit existence.
If limx→af(x)=Llimx→af(x)=L, then for x near a, f(x) is near
A L
B a
C 0
D Infinity
The meaning of limit is that as x approaches a (not necessarily equal), f(x) approaches a fixed number L. Values of f(x) get close to L.
Standard limit limx→0sinxxlimx→0xsinx equals
A 0
B Infinity
C 1
D −1
One of the most basic standard limits is limx→0sinxx=1limx→0xsinx=1 (with x in radians). It is used to evaluate many trig limits.
Standard limit limx→01−cosxx2limx→0x21−cosx equals
A 1
B 0
C 2
D 1/2
Using the standard expansion cosx≈1−x22cosx≈1−2×2 near 0, we get 1−cosxx2≈x2/2×2=1/2×21−cosx≈x2x2/2=1/2.
If limx→af(x)limx→af(x) exists, then
A LHL ≠ RHL
B LHL = RHL
C f(a) must exist
D f(a)=0
Existence of two-sided limit requires equality of left-hand and right-hand limits. f(a) may be undefined or different and still the limit can exist.
Continuity at x=a requires
A All three
B Only limit exists
C f(a) defined
D limit equals f(a)
A function is continuous at a when f(a) exists, limx→af(x)limx→af(x) exists, and both are equal. Missing any one condition breaks continuity.
If f(a) exists but limx→af(x)limx→af(x) does not, then f is
A Continuous
B Differentiable
C Constant
D Discontinuous
Continuity needs both the limit and f(a) and their equality. If the limit does not exist, continuity fails even if the function value exists.
A removable discontinuity happens when
A Limit does not exist
B Limit exists, value differs/undefined
C Limit is infinite
D Graph oscillates
In removable discontinuity, the two-sided limit exists but the function value is either missing or not equal to the limit. Redefining f(a) can “remove” it.
A jump discontinuity means
A Limit infinite
B Function not defined
C Derivative zero
D LHL and RHL unequal
A jump discontinuity occurs when both one-sided limits exist but are not equal. The graph shows a sudden jump at that point.
Infinite discontinuity occurs when
A LHL = RHL
B Value equals limit
C Limit is infinite
D Function constant
If as x approaches a, f(x) grows without bound (±∞), the discontinuity is infinite. This often happens near vertical asymptotes.
Oscillatory discontinuity is common in
A Polynomials
B Step functions
C Constant functions
D sin(1/x)sin(1/x) type
Functions like sin(1/x)sin(1/x) near x=0 oscillate rapidly and do not approach a single value, so the limit fails due to oscillation.
If a function is differentiable at a point, it is always
A Continuous
B Discontinuous
C Periodic
D Constant
Differentiability implies continuity. If the derivative exists at x=a, then the limit equals the function value there, so the function must be continuous at a.
A function can be continuous but not differentiable at
A Smooth point
B Corner point
C Constant region
D Polynomial point
At a corner, left and right derivatives are not equal, so derivative doesn’t exist, but the function may still be unbroken and continuous, like |x| at x=0.
|x| is non-differentiable at
A x=1
B x=2
C x=−2
D x=0
For |x|, the left derivative at 0 is −1 and the right derivative is +1. Since they differ, the derivative at 0 does not exist, though |x| is continuous.
Non-differentiability due to sharp point is called
A Jump
B Removable
C Corner
D Infinite
A corner point has a sharp change in slope where left and right derivatives are finite but unequal. This causes non-differentiability while continuity can still hold.
A cusp typically has slopes
A Infinite opposite
B Equal finite
C Unequal finite
D Always zero
At a cusp, the curve approaches with very steep slopes; left and right slopes often tend to ±∞ (or behave differently), making derivative undefined though the curve may be continuous.
A vertical tangent means derivative is
A 0
B 1
C Undefined constant
D Infinite
A vertical tangent corresponds to slope becoming unbounded (tending to ±∞). Since derivative represents slope, such a point usually has an infinite derivative.
Derivative from first principles is based on
A Product rule
B Integration
C Difference quotient
D Chain rule
First principles use f′(x)=limh→0f(x+h)−f(x)hf′(x)=limh→0hf(x+h)−f(x). It directly measures the limiting slope of the secant line.
For differentiability at x=a, we must have
A Left derivative equals right derivative
B Only LHL exists
C f(a)=0
D Limit infinite
A derivative exists at a point only if both one-sided derivatives exist and are equal. If they differ, the function is not differentiable there.
The geometric meaning of derivative is
A Area under curve
B Slope of tangent
C Length of curve
D Intercept only
The derivative at x=a gives the slope of the tangent line to the curve y=f(x) at that point. It measures instantaneous rate of change.
Second derivative represents
A Total area
B Function value
C Constant term
D Rate of slope change
The second derivative is the derivative of the first derivative. It tells how quickly the slope is changing and is linked to concavity and acceleration ideas.
If f′(x) is constant, then f(x) is
A Linear
B Quadratic
C Cubic
D Periodic
A constant derivative means constant slope. A function with constant slope is a straight line, so f(x) must be linear.
Derivative of a constant is
A 1
B Constant itself
C 0
D Undefined
A constant does not change as x changes, so its rate of change is zero. Hence the derivative of any constant is 0.
Derivative of xnxn is
A xn+1xn+1
B nxnx
C xn−1xn−1
D nxn−1nxn−1
The power rule states ddx(xn)=nxn−1dxd(xn)=nxn−1 for real n (where defined). It is a fundamental differentiation formula.
Product rule is used for derivative of
A f·g
B f+g
C f/g only
D Constant only
When a function is a product of two functions, (fg)′=f′g+fg′(fg)′=f′g+fg′. This rule avoids expanding complicated products and ensures correct differentiation.
Quotient rule applies to
A f·g
B f/g
C f+g
D Constant
For a quotient fggf, the derivative is (fg)′=f′g−fg′g2(gf)′=g2f′g−fg′, assuming g(x) ≠ 0.
Chain rule is used for
A Constant functions
B Only polynomials
C Only linear
D Composite functions
For a composite function f(g(x)), chain rule gives ddxf(g(x))=f′(g(x))⋅g′(x)dxdf(g(x))=f′(g(x))⋅g′(x). It’s essential for nested expressions.
Implicit differentiation is used when
A y is explicit
B function constant
C y given indirectly
D x fixed
If an equation relates x and y together (like x²+y²=1) without solving for y, we differentiate both sides and treat y as a function of x.
Logarithmic differentiation helps in
A Complicated products/powers
B Very simple sums
C Only linear graphs
D Only constants
Taking log simplifies expressions with products, quotients, and variable powers. After differentiating the log form, we convert back, reducing algebra complexity.
Indeterminate form 0/0 suggests
A Limit always 0
B Limit always 1
C Limit infinite
D Needs simplification
The form 0/0 does not determine the limit. We must simplify using factorization, rationalization, standard limits, or L’Hospital’s rule (when applicable).
Indeterminate form ∞/∞ means
A Needs further work
B Limit fixed
C Always ∞
D Always 0
Like 0/0, ∞/∞ does not give a definite value. We simplify by dividing by highest power, using algebraic tricks, or applying L’Hospital’s rule where valid.
0·∞ is an indeterminate form because
A Always 0
B Always ∞
C Needs rewriting
D Not a limit
The product of something tending to 0 and something tending to ∞ can approach many values. We rewrite it as a quotient (like 0/0 or ∞/∞) to evaluate.
∞ − ∞ is indeterminate because
A Always ∞
B Always 0
C Never appears
D Needs common form
Two large terms may cancel partially or fully, giving different results. We combine terms using common denominators or algebraic identities to convert it to a solvable form.
Form 1^∞ often leads to
A 0
B e-type limit
C Always 1
D Always ∞
In 1^∞ type limits, small changes in the base raised to large powers can approach e or other numbers. We typically use logs to convert to exponential form.
L’Hospital rule is mainly used for
A 0/0 or ∞/∞
B 0·∞ only
C All limits
D Polynomials only
L’Hospital’s rule applies to limits in 0/0 or ∞/∞ form (with conditions). We differentiate numerator and denominator and re-check the resulting limit.
Squeeze theorem is useful when
A Function linear
B Derivative asked
C Integral asked
D Direct limit hard
If a function is trapped between two functions with the same limit, then it has the same limit. This helps especially with trig and absolute value expressions.
Continuity of sum of functions requires
A One discontinuous
B Both continuous
C Only one continuous
D No condition
If f and g are continuous at a point, then f+g is also continuous there. Continuity is preserved under addition, subtraction, multiplication, and composition.
Continuity of product requires
A Only one continuous
B Both discontinuous
C Always continuous
D Both continuous
If f and g are continuous at x=a, then fg is continuous at a. This is part of algebra of continuous functions.
Composite function f(g(x)) is continuous if
A g continuous and f at g(a)
B f continuous, g any
C g continuous, f any
D Both discontinuous
For f(g(x)) to be continuous at a, g must be continuous at a and f must be continuous at g(a). Then the composite remains continuous.
Limit at infinity studies behavior as x approaches
A 0
B a
C ∞
D −a
Limits at infinity describe what happens to f(x) when x becomes very large. It helps understand end behavior, horizontal asymptotes, and growth comparison basics.
For rational functions, discontinuity often occurs when
A Numerator zero
B Derivative zero
C Value positive
D Denominator zero
A rational function p(x)q(x)q(x)p(x) is undefined where q(x)=0. Those points can create removable discontinuities (if factors cancel) or infinite discontinuities.
A step function usually shows
A Removable breaks
B Jump breaks
C Smooth curve
D Cusp always
Step functions change value suddenly at certain points. This creates jumps where left and right limits differ, giving jump discontinuities.
One-sided continuity at endpoint a means
A Only inside interval limit
B Two-sided limit exists
C Derivative must exist
D Function constant
At an endpoint, we check continuity using the limit from within the interval only. For left endpoint use RHL; for right endpoint use LHL, along with f(a).
Successive differentiation means finding
A Only first derivative
B Only limits
C Only integrals
D Higher derivatives
Successive differentiation is differentiating repeatedly to get second, third, or nth derivatives. It helps in curvature, Taylor expansions, and patterns in derivatives.
Leibnitz theorem is for nth derivative of
A Sum
B Quotient only
C Product
D Constant
Leibnitz’s theorem gives the nth derivative of a product of two functions using binomial coefficients. It generalizes the product rule to higher derivatives.
In Leibnitz formula, coefficients are
A Binomial coefficients
B Prime numbers
C Fibonacci numbers
D Random constants
Leibnitz’s theorem uses combinations (nk)(kn) in the sum. These binomial coefficients count the ways derivatives split between the two functions.
A function may have limit at a point even if
A f(a) exists
B f(a)=limit
C derivative exists
D f(a) undefined
The limit depends on values near the point, not necessarily at the point. So a hole at x=a (undefined f(a)) can still have a well-defined limit.
If limx→af(x)=f(a)limx→af(x)=f(a), then f is
A Differentiable
B Continuous
C Discontinuous
D Oscillatory
This equality is the key condition for continuity (along with existence of both). When the limit at a equals the function value, the graph has no break there.
A common method for trig limits near 0 is
A Standard limits
B Random substitution
C Only graphs
D Only calculator
Many basic trigonometric limits are evaluated using standard results like sinx/xsinx/x and (1−cosx)/x2(1−cosx)/x2, along with algebraic simplification and identities.