Limits

Limits

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

The limit of a function f(x) as x approaches a is the unique value L that f(x) gets arbitrarily close to, written lim(x→a) f(x) = L. A limit exists only when the left-hand limit (LHL) equals the right-hand limit (RHL) at the point in question. The two standard trigonometric limits every CUET UG student must memorise are lim(x→0) sin(x)/x = 1 and lim(x→0) (1 − cos x)/x = 0, plus the exponential standard lim(x→0) (1 + x)^(1/x) = e. When direct substitution yields 0/0, ∞/∞, 1^∞, or 0^0, the form is indeterminate and needs algebraic manipulation (factorisation, rationalisation) or L’Hôpital’s rule (differentiate numerator and denominator separately). High-yield CUET pointers: (1) For rational functions in x→a form, factorise to cancel the (x − a) term before substituting; (2) confirm LHL = RHL for piecewise functions; (3) x in limit expressions is in radians, never degrees.

---

🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Definition and Existence

Formally, lim(x→a) f(x) = L means: for every ε > 0 there exists a δ > 0 such that |f(x) − L| < ε whenever 0 < |x − a| < δ (the ε–δ definition). A limit is said to exist at x = a only when LHL = lim(x→a⁻) f(x) equals RHL = lim(x→a⁺) f(x). The actual value f(a) is irrelevant for the limit’s existence — removable discontinuities are a classic illustration.

Algebra of Limits

If lim(x→a) f(x) = L and lim(x→a) g(x) = M with M ≠ 0 for division, then:

• Sum/Difference: lim [f(x) ± g(x)] = L ± M
• Product: lim [f(x) · g(x)] = L · M
• Quotient: lim [f(x)/g(x)] = L/M
• Scalar multiple: lim [k · f(x)] = kL

These rules break down when the resulting form is indeterminate: 0/0, ∞/∞, ∞ − ∞, 0·∞, 1^∞, 0^0, ∞^0.

Standard Limits (Memorise)

Limit	Value
lim(x→0) sin x / x	1
lim(x→0) (1 − cos x) / x	0
lim(x→0) (1 − cos x) / x²	1/2
lim(x→0) tan x / x	1
lim(x→0) (1 + x)^(1/x)	e
lim(x→±∞) (1 + a/x)^x	e^a
lim(x→∞) (1 + 1/n)^n	e

Evaluation Techniques

1. Direct substitution — works only when f(a) is defined and the form is not indeterminate.
2. Factorisation — for 0/0 rational functions, factor out (x − a) and cancel before substituting.
3. Rationalisation — multiply numerator and denominator by the conjugate to remove square roots.
4. Substitution method — let t = g(x) so the expression reduces to a standard limit.
5. L’Hôpital’s rule — for 0/0 or ∞/∞, lim f(x)/g(x) = lim f’(x)/g’(x), provided the latter exists. Apply only to true indeterminate forms.
6. Series expansion — replace sin x, cos x, e^x, ln(1+x) by their Maclaurin equivalents near x = 0.

Continuity Link

f is continuous at x = a iff f(a) is defined, lim(x→a) f(x) exists, and lim(x→a) f(x) = f(a). This three-part check is a common CUET MCQ trap: a function may have a finite limit yet fail continuity if f(a) is undefined or differs from L.

CUET Question Patterns

Expect 2–3 MCQs worth 5 marks each, typically: evaluating a rational limit (factorise and substitute), computing a standard trigonometric limit after a substitution, applying L’Hôpital’s once, or testing LHL = RHL on a piecewise function.

---

🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Edge Cases and Subtle Behaviours

Limits of oscillatory functions like lim(x→0) sin(1/x) do not exist because the function oscillates between −1 and +1 without settling — a frequent trap that students wrongly resolve to 0. Similarly, lim(x→0) x·sin(1/x) = 0 because the bounded factor sin(1/x) gets squeezed by the vanishing factor x (sandwich/squeeze theorem: if g(x) ≤ f(x) ≤ h(x) and both g, h tend to L, then f tends to L). Limits at infinity describe end behaviour and identify horizontal asymptotes: lim(x→∞) f(x) = L means y = L is the asymptote.

Indeterminate Form Resolutions

• 0/0 rationals → factor and cancel.
• ∞/∞ → divide numerator and denominator by the highest power of x, or apply L’Hôpital.
• ∞ − ∞ → combine over a common denominator or use algebraic identities (e.g., conjugate multiplication).
• 1^∞ → rewrite as lim [1 + (f(x) − 1)]^(1/(f(x)−1))^(f(x)−1) and apply lim(1 + u)^(1/u) = e; the exponent collapses to lim(f(x) − 1)·g(x), giving e^L.
• 0^0, ∞^0 → take logarithms: let y = f(x)^g(x), then ln y = g(x)·ln f(x); evaluate the resulting 0·∞ or ∞·0 form.

Connection to Differentiation

The derivative is itself defined as a limit: f’(a) = lim(h→0) [f(a + h) − f(a)] / h. Hence every differentiation formula is ultimately justified by a limit argument, and many CUET UG problems in the Calculus section start by asking for the limit that defines a derivative, then asking for f’(a).

Common Mistakes to Avoid

• Confusing lim(x→0) sin(x)/x = 1 (radians) with the degree-mode value π/180 ≈ 0.01745 — CUET assumes radians.
• Applying L’Hôpital’s rule to non-indeterminate forms like 0/1 or ∞/1, which yield 0 and ∞ directly.
• Forgetting to verify LHL = RHL for piecewise-defined functions such as f(x) = x for x < 1 and 2x − 1 for x ≥ 1, where both one-sided limits equal 1 but the function itself is continuous only at the matched value.
• Treating ∞ − ∞ as 0 instead of as an indeterminate form requiring algebraic rewriting.
• Dropping the exponent when using lim(1 + a/x)^x = e^a, mistakenly writing the limit as 1 instead of e^a.

Worked Example

Evaluate lim(x→0) (sin 3x) / (sin 5x). Substitute t = 3x; then as x → 0, t → 0 and sin 5x = sin(5t/3). The expression becomes (sin t)/(sin(5t/3)) · 1 = (sin t / t) · (5t/3) / (sin(5t/3)) · 3/5 → 1 · 1 · 3/5 = 3/5.

Practice Prompts

1. Evaluate lim(x→2) (x² − 4) / (x − 2) using factorisation, and state why the limit exists even though f(2) is undefined.
2. Compute lim(x→0) (e^(2x) − 1) / (sin 5x) by applying L’Hôpital’s rule once and verify the answer using the standard limits table.

---

Content adapted based on your selected roadmap duration. Switch tiers using the selector above.