📝 One sided vs two sided limits (14 MCQs)
📖 From Calculus • 2. Limits and Continuity an Introduction • 14 questions available
What is One sided vs two sided limits?
Definition:
A two-sided limit exists if and only if both the left-hand limit and the right-hand limit exist and are equal to the same value . One-sided limits are independent evaluations from each direction, while the two-sided limit requires agreement from both sides, ensuring the function approaches the same value regardless of the approach path.
Example:
For , check .
Solution: , . Since they differ, the two-sided limit does not exist.
Reason:
Distinguishing these limits is vital for continuity analysis and for understanding functions with corners or jumps, as the existence of the two-sided limit is a prerequisite for differentiability and integrability at a point.
📝 All One sided vs two sided limits MCQs
Q1. What condition must be satisfied for the two‑sided limit \\\displaystyle\\lim_{x\\to a}f(x)\ to exist?
📖 Explanation: The two‑sided limit exists precisely when the left‑hand and right‑hand limits both exist and approach the same number L. Equality of the one‑sided limits ensures that approaching a from either direction yields the same value, which is the definition of the overall limit.
Q2. For the function \f(x)=\\frac{|x|}{x}\, what is \\\displaystyle\\lim_{x\\to 0^-}f(x)\?
📖 Explanation: When x approaches 0 from the left, |x| equals -x, so \\\frac{|x|}{x}=\\frac{-x}{x}=-1\. This value is constant for all negative x, therefore the left‑hand limit as x tends to 0 is -1. The limit does not depend on any other behavior of the function.
Q3. If \\\displaystyle\\lim_{x\\to a^-}f(x)=2\ and \\\displaystyle\\lim_{x\\to a^+}f(x)=3\, which statement is a valid inference?
📖 Explanation: Since the left‑hand limit approaches 2 while the right‑hand limit approaches 3, the two approaches give different values. By definition, a two‑sided limit exists only when both one‑sided limits exist and are equal. Hence the overall limit at a does not exist.
Q4. Suppose \\\displaystyle\\lim_{x\\to a^-}f(x)=\\displaystyle\\lim_{x\\to a^+}f(x)=L\. Which of the following must be true?
📖 Explanation: The equality of the left‑hand and right‑hand limits guarantees that each one‑sided limit exists and they share the same value L. Continuity further requires that the function's actual value at a equals L, which is not imposed by the limit condition alone. Thus only the existence and equality of the one‑sided limits are certain.
Q5. Changing the value of \f\ at the single point \x=a\ will affect which of the following limits?
📖 Explanation: Limits depend on the behavior of f arbitrarily close to a, but not on the function's value exactly at a. Modifying f(a) changes the pointwise definition without influencing the approach from either side, so neither the left‑hand nor the right‑hand limit is altered.
Q6. A piecewise function is defined by \f(x)=\\begin{cases}1,&x
📖 Explanation: Approaching a from the left yields the constant value 1, while approaching from the right yields the constant value 3. Because the two one‑sided limits are different, the definition of a two‑sided limit fails, so the limit at a does not exist.
Q7. Two graphs differ only by the point at \x=a\. How does this affect the limits as \x\\to a\?
📖 Explanation: Limits are determined by values arbitrarily close to the point of interest, not by the function's value at the point itself. Changing the function at a single point does not alter the approach from either side, so all one‑sided and two‑sided limits remain unchanged.
Q8. Let \f(x)=\\begin{cases}x^{2},&x<0\\\\2x,&x\\ge0\\end{cases}\. Find \\\displaystyle\\lim_{x\\to 0^-}f(x)\ and \\\displaystyle\\lim_{x\\to 0^+}f(x)\.
📖 Explanation: For x approaching 0 from the left, f(x)=x^{2} tends to 0 because x^{2}\\to0. From the right, f(x)=2x also tends to 0 as x\\to0^{+}. Both one‑sided limits equal 0, establishing the values requested.
Q9. Which statement correctly describes the meaning of the symbol \+\\infty\ in a limit expression?
📖 Explanation: The notation \+\\infty\ indicates that as the variable approaches the limiting point, the function's values increase without any finite upper bound. It is not a real number but a way to express unbounded growth in the positive direction.
Q10. Consider \g(x)=\\sin\\frac{1}{x}\ for \x\\neq0\. Why does \\\displaystyle\\lim_{x\\to0}g(x)\ not exist, even though both one‑sided limits are undefined?
📖 Explanation: As x approaches 0, the argument \1/x\ grows without bound, causing \\\sin(1/x)\ to swing between -1 and 1 infinitely many times. No single number can capture this behavior, so the overall limit fails to exist despite the lack of distinct one‑sided limits.
Q11. If \\\displaystyle\\lim_{x\\to a}f(x)=L\, what must be true for every sequence \\\{x_n\\}\ with \x_n\\neq a\ and \x_n\\to a\?
📖 Explanation: The ε‑δ definition of a limit is equivalent to the sequential criterion: for any sequence approaching a (excluding a itself), the corresponding function values must converge to the same limit L. Hence every such sequence of f(x_n) must tend to L.
Q12. Construct a function \h\ such that \\\displaystyle\\lim_{x\\to 1^-}h(x)=\\displaystyle\\lim_{x\\to 1^+}h(x)=4\ but \h\ is discontinuous at \x=1\. Which of the following definitions achieves this?
📖 Explanation: Defining \h(x)=4\ everywhere except at the point x=1, where we assign a different value (e.g., 0), preserves the left‑hand and right‑hand limits of 4 while creating a removable discontinuity at x=1. The limits remain 4, but the function value at the point does not match, breaking continuity.
Q13. How does the ε‑δ definition of \\\displaystyle\\lim_{x\\to a}f(x)=L\ guarantee the equality of the left‑hand and right‑hand limits?
📖 Explanation: The ε‑δ definition states that for every ε>0 there exists a δ>0 such that whenever 0<|x−a|<δ, the inequality |f(x)−L|<ε holds. This condition applies to all x within the δ‑neighborhood, regardless of whether x is less than or greater than a, thereby forcing both one‑sided limits to equal L.
Q14. A function \p\ has a removable discontinuity at \x=c\ where \\\displaystyle\\lim_{x\\to c}p(x)=5\ but \p(c)=2\. After redefining \p(c)=5\, which statement is true?
📖 Explanation: By assigning the function's value at the point of discontinuity to match the existing limit (5), the gap is removed. The limit remains 5, and now the function's value equals the limit, satisfying the definition of continuity at \c\. Hence the function is continuous there.