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Indietro

Calculus: Limits

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  • What is the definition of a limit in calculus?

    The limit of a function \(f(x)\) as \(x\) approaches a value \(a\) is the value that \(f(x)\) gets closer to as \(x\) approaches \(a\).

  • How is the limit notation written?

    The limit of \(f(x)\) as \(x\) approaches \(a\) is written as \(\lim_{x \to a} f(x)\).

  • What does it mean if \(\lim_{x \to a} f(x) = L\)?

    It means that as \(x\) gets arbitrarily close to \(a\), the function values \(f(x)\) approach the number \(L\).

  • What is a one-sided limit?

    A one-sided limit considers the behavior of \(f(x)\) as \(x\) approaches \(a\) from only one side: left (\(x \to a^-\)) or right (\(x \to a^+\)).

  • When does the limit \(\lim_{x \to a} f(x)\) exist?

    The limit exists if and only if the left-hand limit and right-hand limit at \(a\) are equal.

  • What is the limit of a constant function \(f(x) = c\) as \(x \to a\)?

    The limit is the constant itself: \(\lim_{x \to a} c = c\).

  • How do you find the limit of a polynomial function as \(x \to a\)?

    For polynomials, the limit is the value of the polynomial evaluated at \(a\): \(\lim_{x \to a} P(x) = P(a)\).

  • What is the limit of \(\frac{1}{x}\) as \(x \to 0^+\)?

    The limit is positive infinity because values approach very large positive numbers.

  • What is the limit of \(\frac{1}{x}\) as \(x \to 0^-\)?

    The limit is negative infinity because values approach very large negative numbers.

  • What is an indeterminate form in limits?

    An indeterminate form occurs when direct substitution results in expressions like \(\frac{0}{0}\) or \(\infty - \infty\), requiring further analysis.

  • What is the Squeeze Theorem?

    If \(f(x) \leq g(x) \leq h(x)\) near \(a\) and \(\lim_{x \to a} f(x) = \lim_{x \to a} h(x) = L\), then \(\lim_{x \to a} g(x) = L\).

  • How do you evaluate limits involving infinity?

    Analyze the dominant terms in numerator and denominator to determine if the limit approaches infinity, zero, or a finite number.

  • What is the limit of \(\frac{1}{x^2}\) as \(x \to 0\)?

    The limit is infinity because values become very large positive numbers from both sides.

  • What is the difference between continuity and limits?

    Continuity at a point means the limit exists and equals the function value; limits alone describe approaching behavior.

  • What is the formal epsilon-delta definition of a limit?

    For every \(\varepsilon > 0\), there exists \(\delta > 0\) such that if \(0 < |x - a| < \delta\), then \(|f(x) - L| < \varepsilon\).

  • What is a removable discontinuity in terms of limits?

    A point where the limit exists but the function is not defined or differs from the limit value.

  • How do limits help define the derivative?

    The derivative is defined as \(f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}\).

  • What is the limit of \(\sin x / x\) as \(x \to 0\)?

    The limit is 1.

  • What is the limit of \((1 + \frac{1}{n})^n\) as \(n \to \infty\)?

    The limit is the number e, approximately 2.718.

  • What is the limit of \(\sqrt{x^2 + 1} - x\) as \(x \to \infty\)?

    The limit is 0, found by rationalizing the expression.