JEE · Mathematics
Understand one-sided and two-sided limits, resolve indeterminate forms with valid methods, and test continuity by distinguishing the function value from the value approached near the point.
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In short
A limit describes what f(x) approaches as x approaches a point; it does not require f(a) to equal that value or even to be defined.
A two-sided real limit exists only when the left-hand and right-hand limits exist and are equal. A function is continuous at a when f(a) is defined and lim[x->a]f(x)=f(a).
The official JEE documents define content scope. They do not publish chapter weightage, so none is asserted here.
JEE Main 2026 and JEE Advanced 2026 are independent authorities.
A limit does not require f(a) to equal that value or even to be defined.
A limit describes what f(x) approaches as x approaches a point. It does not require f(a) to equal that value or even to be defined at all.
A two-sided real limit exists only when the left-hand and right-hand limits exist and are equal.
For piecewise or absolute-value functions especially, check the left-hand limit and right-hand limit separately before claiming the two-sided limit exists.
Substitution finishes the limit only in a continuous algebraic context; an indeterminate form requires transformation first.
Try substitution first. If it produces an ordinary finite value in a continuous algebraic context, that may finish the limit. If it produces 0/0, infinity/infinity or another indeterminate form, transform the expression before computing.
Continuity at a requires f(a) defined and lim[x->a]f(x)=f(a).
Continuity is established by checking the function value, the left-hand limit and the right-hand limit as three separate quantities and confirming all three agree.
L'Hospital rule applies only to an eligible 0/0 or infinity/infinity limit under its stated hypotheses.
L'Hospital rule replaces f/g by f'/g' for a valid 0/0 or infinity/infinity limit, provided the functions are differentiable on an appropriate deleted neighbourhood, the denominator derivative is nonzero there where required, the original form is eligible, and the derivative quotient has a limit under the theorem conditions. It is not an explicit JEE Main 2026 bullet.
If f is continuous on [a,b] and N lies between f(a) and f(b), some c in [a,b] satisfies f(c)=N.
The intermediate value property requires real continuity on a closed interval. It is explicit in JEE Advanced 2026 and is not usable for discontinuous functions.
Match the limit or continuity signal to the correct first method.
Let f(x)=(x²-1)/(x-1) for x≠1, and f(1)=k. Find k for continuity at x=1.
Direct substitution into the fraction gives 0/0, so it is indeterminate. Factor for x≠1: (x²-1)/(x-1)=(x-1)(x+1)/(x-1)=x+1. Therefore lim[x->1]f(x)=2.
Continuity requires f(1)=2, so k=2. Validity check: LHL=RHL=2 and the repaired function value is also 2.
Advanced records are explicit JEE Advanced 2026 scope, not JEE Main 2026 scope.
Calling 0/0 the value of a limit.
Knowledge gapWhy it happens
0/0 is an indeterminate form, not a computed value; it signals that transformation is required.
How it is corrected
Transform the expression by factoring, rationalising or another valid method before computing the limit.
Assuming the limit equals f(a).
Knowledge gapWhy it happens
A limit does not require f(a) to equal that value or even to be defined.
How it is corrected
Check f(a), the left-hand limit and the right-hand limit as three separate quantities.
Cancelling a factor without remembering that cancellation is valid only away from the cancelled zero.
Execution errorWhy it happens
The simplified expression and the original expression differ exactly at the cancelled point.
How it is corrected
Cancel only for the punctured expression, then take the limit of the simplified form.
Ignoring one-sided limits for piecewise or absolute-value functions.
Decision / selection errorWhy it happens
A two-sided limit exists only when both one-sided limits exist and agree.
How it is corrected
Compute the left-hand and right-hand limits separately before claiming the two-sided limit exists.
Using degree-based reasoning for standard trigonometric small-angle limits.
Execution errorWhy it happens
The standard trig limits such as lim[x->0] sinx/x=1 require x in radians.
How it is corrected
Confirm the angle is in radians before applying a standard trig limit.
Using L'Hospital rule in Main preparation as though it were an explicit Main 2026 bullet.
Knowledge gapWhy it happens
L'Hospital rule is explicit in JEE Advanced 2026 but is not explicitly named in JEE Main 2026 Unit 7.
How it is corrected
Check the official-syllabus mapping table on this page before treating L'Hospital as Main-scope.
Using L'Hospital rule on a noneligible form without transformation.
Decision / selection errorWhy it happens
L'Hospital rule applies to an eligible 0/0 or infinity/infinity limit under its stated hypotheses, not to 0·infinity or infinity-infinity directly.
How it is corrected
Verify the indeterminate form and every hypothesis before applying L'Hospital rule.
Assuming continuity implies differentiability.
Knowledge gapWhy it happens
Continuity and differentiability are separate properties; continuity does not by itself guarantee differentiability.
How it is corrected
Check the derivative definition separately before claiming differentiability from continuity alone.
FAQ
Straight answers about how Rank Sarthi fits into serious exam preparation.
It is an indeterminate form that does not determine the limit by itself; the expression must be transformed, for example by factoring or rationalising, before the limit can be computed.
Only when the left-hand limit and the right-hand limit both exist and are equal.
f(a) must be defined, the limit as x approaches a must exist, and that limit must equal f(a).
No. JEE Main 2026 Unit 7 does not explicitly name L'Hospital rule. It is explicit in JEE Advanced 2026.
Only for an eligible 0/0 or infinity/infinity limit, after verifying differentiability on an appropriate deleted neighbourhood and the other stated hypotheses.
Radian-based trig limits, quotient denominator conditions, one-sided existence and L'Hospital hypotheses must be checked against source before publication. Do not publish counts or trends.
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