Question 1

What is a limit in calculus?

Accepted Answer

A limit describes the value that a function approaches as the input (x) approaches a particular value. Written as lim(x→a) f(x) = L, it means as x gets arbitrarily close to a, f(x) gets arbitrarily close to L. Limits are fundamental to calculus, forming the basis for derivatives and integrals. A limit can exist even if the function is undefined at that point, which is why limits are more general than simple function evaluation.

Question 2

How do you calculate a limit?

Accepted Answer

To calculate a limit: 1) Try direct substitution - if the function is continuous at the point, the limit equals f(a). 2) If you get 0/0 or ∞/∞ (indeterminate forms), try factoring and canceling, rationalizing, or using L'Hôpital's rule. 3) For limits at infinity, analyze the highest degree terms. 4) For piecewise functions, check left and right limits separately. 5) Use limit laws to break complex limits into simpler parts.

Question 3

What is the difference between left-hand and right-hand limits?

Accepted Answer

A left-hand limit (x→a⁻) approaches from values less than a (from the left on a number line). A right-hand limit (x→a⁺) approaches from values greater than a (from the right). For a limit to exist at a point, both one-sided limits must exist and be equal: lim(x→a⁻) f(x) = lim(x→a⁺) f(x) = lim(x→a) f(x). If they differ, the two-sided limit does not exist, indicating a jump discontinuity.

Question 4

What are indeterminate forms in limits?

Accepted Answer

Indeterminate forms arise when direct substitution doesn't give a definite answer. Common forms: 0/0 (most common), ∞/∞, 0×∞, ∞-∞, 0⁰, 1^∞, ∞⁰. These require special techniques: factoring and canceling, rationalizing, L'Hôpital's rule (take derivatives of numerator and denominator), or series expansion. For example, lim(x→0) (sin x)/x = 0/0, but the actual limit is 1.

Question 5

What is L'Hôpital's rule?

Accepted Answer

L'Hôpital's rule states that if lim(x→a) f(x)/g(x) gives 0/0 or ∞/∞, then lim(x→a) f(x)/g(x) = lim(x→a) f'(x)/g'(x), provided the limit of derivatives exists. You take derivatives of the numerator and denominator separately (not the quotient rule). Example: lim(x→0) (sin x)/x = lim(x→0) (cos x)/1 = 1. You can apply L'Hôpital's rule multiple times if needed.

Question 6

How do you find limits at infinity?

Accepted Answer

For limits at infinity: 1) For polynomials, the limit is determined by the highest degree term. 2) For rational functions, compare degrees of numerator and denominator. If degrees are equal, limit = ratio of leading coefficients. If numerator degree > denominator, limit = ±∞. If denominator degree > numerator, limit = 0. 3) For exponentials, consider the base and exponent. 4) Divide by highest power of x to simplify.

Question 7

What does it mean when a limit does not exist?

Accepted Answer

A limit does not exist (DNE) when: 1) Left and right limits are different (jump discontinuity). 2) Function oscillates infinitely near the point (like sin(1/x) as x→0). 3) Function approaches different infinities from each side. 4) Function is unbounded and doesn't settle. Example: lim(x→0) 1/x DNE because left limit = -∞ and right limit = +∞. However, limits can be infinite (like lim(x→0) 1/x² = ∞) and still 'exist' in the extended real numbers.

Question 8

What is direct substitution in limits?

Accepted Answer

Direct substitution means plugging the value directly into the function: lim(x→a) f(x) = f(a). This works when the function is continuous at x = a. For example, lim(x→2) (x² + 1) = 2² + 1 = 5. If direct substitution gives a determinate value (not 0/0 or ∞/∞), that's the limit. Polynomial, exponential, and trigonometric functions are continuous everywhere (or almost everywhere), making direct substitution often applicable.

Question 9

How do you use limit laws to calculate limits?

Accepted Answer

Limit laws allow you to break complex limits into simpler parts: Sum rule: lim[f(x) + g(x)] = lim f(x) + lim g(x). Product rule: lim[f(x) × g(x)] = lim f(x) × lim g(x). Quotient rule: lim[f(x)/g(x)] = lim f(x) / lim g(x) if lim g(x) ≠ 0. Constant multiple: lim[c × f(x)] = c × lim f(x). Power rule: lim[f(x)]ⁿ = [lim f(x)]ⁿ. These laws only apply when individual limits exist.

Question 10

What are limits used for in calculus?

Accepted Answer

Limits are the foundation of calculus and define: 1) Derivatives: f'(x) = lim(h→0) [f(x+h) - f(x)]/h measures instantaneous rate of change. 2) Integrals: defined as limits of Riemann sums. 3) Continuity: f is continuous at a if lim(x→a) f(x) = f(a). 4) Series convergence: whether infinite sums have finite values. 5) End behavior: limits at ±∞ describe long-term function behavior. Limits make precise the intuitive notion of 'approaching' a value.

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