A set of questions on the concepts of the limit of a function in calculus are presented along with detailed answers. These questions are designed to help you gain a deep understanding of limits, which is essential for concepts like the derivative and integrals. They also help identify concepts that need review.
True or False: If a function \( f \) is not defined at \( x = a \), then the limit \( \lim_{x \to a} f(x) \) never exists.
Answer: False. The limit \( \lim_{x \to a} f(x) \) may exist even if \( f \) is undefined at \( x = a \), because limits depend on the behavior of \( f \) close to \( a \), not at \( a \).
True or False: If \( f \) and \( g \) are two functions such that \[ \lim_{x \to a} f(x) = +\infty \quad \text{and} \quad \lim_{x \to a} g(x) = +\infty, \] then \[ \lim_{x \to a} [f(x) - g(x)] \] is always 0.
Answer: False. Infinity is not a number, so \( +\infty - \infty \) is undefined. \( +\infty \) and \( -\infty \) are symbols representing very large or very small quantities.
True or False: The graph of a rational function may cross its vertical asymptote.
Answer: False. Vertical asymptotes occur at \( x \)-values that make the denominator 0, where the function is undefined.
True or False: The graph of a function may cross its horizontal asymptote.
Answer: True. For example: \[ f(x) = \frac{x - 2}{(x - 1)(x + 3)} \] The degree of the denominator (2) is higher than the numerator (1), giving a horizontal asymptote \( y = 0 \). However, the x-intercept at \( x = 2 \) crosses the horizontal asymptote.
If \( f(x) \) and \( g(x) \) satisfy \[ \lim_{x \to a} f(x) = +\infty \quad \text{and} \quad \lim_{x \to a} g(x) = 0, \] then which statements about \(\lim_{x \to a} [f(x) \cdot g(x)]\) are true?
Answer: (C) and (D). Examples: \[ f(x) = \frac{1}{x}, \; g(x) = 2x \quad \text{as } x \to 0 \] \[ f(x) = \frac{1}{x^2}, \; g(x) = x \quad \text{as } x \to 0 \]
True or False: If \(\lim_{x \to a} f(x)\) and \(\lim_{x \to a} g(x)\) exist, then \[ \lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}. \]
Answer: False. This holds only if \(\lim_{x \to a} g(x) \neq 0\).
True or False: For any polynomial \( p(x) \), \(\lim_{x \to a} p(x) = p(a)\).
Answer: True. Polynomials are continuous functions.
True or False: If \(\lim_{x \to a^-} f(x) = L_1\) and \(\lim_{x \to a^+} f(x) = L_2\), then \(\lim_{x \to a} f(x)\) exists only if \(L_1 = L_2\).
Answer: True. This is a fundamental property of limits.
True or False: \(\lim_{x \to \infty} \sin x = \pm 1\).
Answer: False. \(\sin x\) oscillates indefinitely and has no limit as \(x \to \infty\) or \(x \to -\infty\). Same for \(\cos x\).
Calculus questions with answers and Calculus tutorials and problems.