# 4.1 MCQs-Limits and Continuity ### Limits and Continuity MCQs ### Concept of Limits 1\. The statement $$\lim\_{x \to a} f(x) = L$$ means that: 1. $$f(a) = L$$ 2. $$f(x)$$ gets arbitrarily close to L as x gets sufficiently close to a (from either side) 3. $$f(x)$$ equals L for all x near a 4. The value of $$f(a)$$ exists
Show me the answer **Answer:** 2. $$f(x)$$ gets arbitrarily close to L as x gets sufficiently close to a (from either side) **Explanation:** * The limit describes the behavior of the function $$f(x)$$ as x **approaches** a, not necessarily the value **at** $$x = a$$. * The function may or may not be defined at $$x = a$$, and even if defined, $$f(a)$$ may not equal L. * The formal ($$\epsilon-\delta$$) definition captures this idea of making $$f(x)$$ arbitrarily close to L by making x sufficiently close to a.
*** 2\. $$\lim\_{x \to 0} \frac{\sin x}{x}$$ is equal to: 1. 0 2. 1 3. Does not exist 4. $$\infty$$
Show me the answer **Answer:** 2. 1 **Explanation:** * This is a standard trigonometric limit. * $$\lim\_{x \to 0} \frac{\sin x}{x} = 1$$ * This result is fundamental and is often proved using the Squeeze Theorem or geometric arguments. * Note that $$\lim\_{x \to 0} \frac{\tan x}{x} = 1$$ and $$\lim\_{x \to 0} \frac{1 - \cos x}{x} = 0$$ are also important.
*** 3\. If $$\lim\_{x \to a^-} f(x) = 5$$ and $$\lim\_{x \to a^+} f(x) = 5$$, then: 1. $$\lim\_{x \to a} f(x) = 5$$ 2. $$f(a) = 5$$ 3. $$f(x)$$ is continuous at $$x = a$$ 4. The limit does not exist
Show me the answer **Answer:** 1. $$\lim\_{x \to a} f(x) = 5$$ **Explanation:** * For the (two-sided) limit $$\lim\_{x \to a} f(x)$$ to exist and equal L, both the left-hand limit ($$x \to a^-$$) and the right-hand limit ($$x \to a^+$$) must exist and be equal to L. * Here, both one-sided limits equal 5, so the two-sided limit exists and equals 5. * This does not guarantee that $$f(a) = 5$$ or that $$f$$ is continuous at a (for continuity, we also need $$f(a)$$ to be defined and equal to this limit).
*** 4\. $$\lim\_{x \to \infty} \frac{1}{x}$$ is equal to: 1. $$\infty$$ 2. 1 3. 0 4. Does not exist
Show me the answer **Answer:** 3. 0 **Explanation:** * As x grows larger and larger (approaches infinity), the value of $$\frac{1}{x}$$ becomes smaller and smaller, approaching 0. * Formally, $$\lim\_{x \to \infty} \frac{1}{x} = 0$$. * Similarly, $$\lim\_{x \to -\infty} \frac{1}{x} = 0$$.
*** ### Limit Laws and Indeterminate Forms 5\. If $$\lim\_{x \to 2} f(x) = 3$$ and $$\lim\_{x \to 2} g(x) = 4$$, then $$\lim\_{x \to 2} \[2f(x) - g(x)]$$ equals: 1. 2 2. 5 3. 7 4. 14
Show me the answer **Answer:** 1. 2 **Explanation:** * Using limit laws (specifically, the difference law and constant multiple law): $$\lim\_{x \to 2} \[2f(x) - g(x)] = 2 \cdot \lim\_{x \to 2} f(x) - \lim\_{x \to 2} g(x)$$ * Substitute the given limits: $$= 2 \cdot 3 - 4 = 6 - 4 = 2$$
*** 6\. Which of the following is an indeterminate form? 1. $$\frac{0}{0}$$ 2. $$\frac{\infty}{\infty}$$ 3. $$0 \cdot \infty$$ 4. All of the above
Show me the answer **Answer:** 4. All of the above **Explanation:** * Indeterminate forms are expressions that do not guarantee a specific limit value without further analysis. Common indeterminate forms include: * $$\frac{0}{0}$$ (e.g., $$\lim\_{x \to 0} \frac{\sin x}{x} = 1$$, but $$\lim\_{x \to 0} \frac{x}{x^2} = \infty$$) * $$\frac{\infty}{\infty}$$ * $$0 \cdot \infty$$ * $$\infty - \infty$$ * $$0^0$$ * $$1^\infty$$ * $$\infty^0$$ * Techniques like factoring, rationalization, or L'Hôpital's Rule are used to evaluate them.
*** 7\. $$\lim\_{x \to 3} \frac{x^2 - 9}{x - 3}$$ is equal to: 1. 0 2. 6 3. Does not exist 4. $$\infty$$
Show me the answer **Answer:** 2. 6 **Explanation:** * Direct substitution gives $$\frac{0}{0}$$, an indeterminate form. * Factor the numerator: $$x^2 - 9 = (x - 3)(x + 3)$$ * Simplify: $$\frac{(x - 3)(x + 3)}{x - 3} = x + 3$$, for $$x \neq 3$$ * Now take the limit: $$\lim\_{x \to 3} (x + 3) = 3 + 3 = 6$$
*** ### Continuity 8\. A function $$f(x)$$ is continuous at a point $$x = a$$ if: 1. $$f(a)$$ is defined 2. $$\lim\_{x \to a} f(x)$$ exists 3. $$\lim\_{x \to a} f(x) = f(a)$$ 4. All of the above
Show me the answer **Answer:** 4. All of the above **Explanation:** * The three conditions for continuity at a point $$x = a$$ are: 1. $$f(a)$$ is defined (the point exists). 2. The limit $$\lim\_{x \to a} f(x)$$ exists. 3. The limit equals the function value: $$\lim\_{x \to a} f(x) = f(a)$$. * If any one of these conditions fails, the function has a discontinuity at $$x = a$$.
*** 9\. The function $$f(x) = \frac{1}{x}$$ is: 1. Continuous everywhere 2. Continuous for all $$x \neq 0$$ 3. Continuous only at $$x = 0$$ 4. Discontinuous everywhere
Show me the answer **Answer:** 2. Continuous for all $$x \neq 0$$ **Explanation:** * The function $$f(x) = \frac{1}{x}$$ is a rational function. Rational functions are continuous at every point in their domain. * The domain of $$f(x) = \frac{1}{x}$$ is all real numbers except $$x = 0$$. * Therefore, it is continuous for all $$x \neq 0$$. * At $$x = 0$$, it is not defined, so it cannot be continuous there. It has an infinite discontinuity (vertical asymptote) at $$x = 0$$.
*** 10\. The greatest integer function $$f(x) = \lfloor x \rfloor$$ (floor function) is discontinuous at: 1. All integer points 2. All rational points 3. All irrational points 4. It is continuous everywhere
Show me the answer **Answer:** 1. All integer points **Explanation:** * The floor function $$\lfloor x \rfloor$$ gives the greatest integer less than or equal to x. * At any integer value, say $$x = n$$: * The left-hand limit: $$\lim\_{x \to n^-} \lfloor x \rfloor = n - 1$$ * The right-hand limit: $$\lim\_{x \to n^+} \lfloor x \rfloor = n$$ * Since the left and right limits are not equal, the two-sided limit does not exist. * Therefore, the function has a **jump discontinuity** at every integer.
*** ### Intermediate Value Theorem (IVT) 11\. The Intermediate Value Theorem states that if a function $$f$$ is continuous on the closed interval $$\[a, b]$$ and $$N$$ is any number between $$f(a)$$ and $$f(b)$$, then: 1. $$f(c) = N$$ for exactly one $$c$$ in $$(a, b)$$ 2. $$f(c) = N$$ for at least one $$c$$ in $$(a, b)$$ 3. $$f(c) = 0$$ for some $$c$$ in $$(a, b)$$ 4. $$f$$ is differentiable on $$(a, b)$$
Show me the answer **Answer:** 2. $$f(c) = N$$ for at least one $$c$$ in $$(a, b)$$ **Explanation:** * The IVT guarantees the existence of **at least one** value $$c$$ in the open interval $$(a, b)$$ such that $$f(c) = N$$. * It does **not** guarantee uniqueness. There could be multiple such $$c$$ values. * A key application is proving that equations have roots. If $$f(a)$$ and $$f(b)$$ have opposite signs, then by taking $$N = 0$$, the theorem guarantees a root in $$(a, b)$$.
*** 12\. Which condition is necessary to apply the Intermediate Value Theorem? 1. The function must be differentiable on $$(a, b)$$ 2. The function must be continuous on $$\[a, b]$$ 3. The function must be one-to-one on $$\[a, b]$$ 4. $$f(a)$$ and $$f(b)$$ must have the same sign
Show me the answer **Answer:** 2. The function must be continuous on $$\[a, b]$$ **Explanation:** * The key hypothesis of the IVT is that the function is **continuous** on the **closed** interval $$\[a, b]$$. * Differentiability is a stronger condition and is not required. * The function does not need to be one-to-one. * For the typical root-finding application, $$f(a)$$ and $$f(b)$$ must have **opposite** signs.
*** ### Limits at Infinity and Asymptotes 13\. If the degree of the numerator is less than the degree of the denominator in a rational function $$R(x) = \frac{P(x)}{Q(x)}$$, then $$\lim\_{x \to \pm \infty} R(x)$$ is: 1. $$\infty$$ 2. The ratio of the leading coefficients 3. 0 4. Does not exist
Show me the answer **Answer:** 3. 0 **Explanation:** * For limits at infinity of rational functions: * If degree(Numerator) < degree(Denominator): The limit is 0. The x-axis ($$y=0$$) is a horizontal asymptote. * If degree(Numerator) = degree(Denominator): The limit is the ratio of the leading coefficients. * If degree(Numerator) > degree(Denominator): The limit is $$\infty$$ or $$-\infty$$ (or DNE), and there is an oblique/slant asymptote.
*** 14\. The line $$x = a$$ is a vertical asymptote of the function $$y = f(x)$$ if: 1. $$\lim\_{x \to a} f(x) = 0$$ 2. $$\lim\_{x \to a} f(x) = \infty$$ or $$-\infty$$ 3. $$\lim\_{x \to \infty} f(x) = a$$ 4. $$f(a) = 0$$
Show me the answer **Answer:** 2. $$\lim\_{x \to a} f(x) = \infty$$ or $$-\infty$$ **Explanation:** * A vertical asymptote occurs at $$x = a$$ if the function's values increase or decrease without bound as x approaches a from the left, right, or both. * Formally, at least one of these one-sided limits is infinite: $$\lim\_{x \to a^-} f(x) = \pm\infty$$ or $$\lim\_{x \to a^+} f(x) = \pm\infty$$. * For rational functions, vertical asymptotes occur at the zeros of the denominator (provided they are not also zeros of the numerator).
*** ### Squeeze Theorem 15\. The Squeeze Theorem is useful for finding limits when: 1. The function is piecewise defined 2. The function can be bounded between two other functions whose limits are known and equal 3. Direct substitution yields an indeterminate form 4. The function is discontinuous
Show me the answer **Answer:** 2. The function can be bounded between two other functions whose limits are known and equal **Explanation:** * The Squeeze Theorem (or Sandwich Theorem) states: If $$g(x) \le f(x) \le h(x)$$ for all x near a (except possibly at a), and $$\lim\_{x \to a} g(x) = \lim\_{x \to a} h(x) = L$$, then $$\lim\_{x \to a} f(x) = L$$. * It is particularly helpful for limits involving trigonometric functions (like $$\lim\_{x \to 0} \frac{\sin x}{x}$$) or oscillating functions multiplied by a decaying function.