Home / AP Calculus BC 1.4 Estimating Limit Values from Tables Study Notes

AP Calculus BC 1.4 Estimating Limit Values from Tables Study Notes

AP Calculus BC 1.4 Estimating Limit Values from Tables Study Notes - New Syllabus

AP Calculus BC 1.4 Estimating Limit Values from Tables Study Notes- New syllabus

AP Calculus BC 1.4 Estimating Limit Values from Tables Study Notes – AP Calculus BC-  per latest AP Calculus BC Syllabus.

LEARNING OBJECTIVE

  • Estimating Limit Values from Tables

Key Concepts: 

  • Estimating Limit Values from Tables

AP Calculus BC-Concise Summary Notes- All Topics

AP Calculus BC Exam Questions MCQs and FRQs

Estimating Limit Values from Tables

Estimating Limit Values from Tables

Limits can be estimated numerically using a table of function values near the point of interest. This is especially useful when a function is too complicated to evaluate analytically.

Key Idea:

  • Choose values of \( x \) that approach the target point \( a \) from both the left (smaller than \( a \)) and the right (larger than \( a \)).
  • Compute \( f(x) \) at those values and observe what number the function values are approaching.
  • If the values from both sides approach the same number, that is the estimated limit.

Important Note: The function value at \( x = a \) does not affect the limit. The limit depends only on values near \( a \).

Example :

Use the table of values to estimate \( \lim_{x \to 2} \frac{x^2 – 4}{x – 2} \).

\( x \)\( f(x) = \frac{x^2 – 4}{x – 2} \)
1.93.9
1.993.99
1.9993.999
2.0014.001
2.014.01
2.14.1
▶️ Answer/Explanation

As \( x \) gets closer to 2 from the left and right, \( f(x) \) approaches 4.

\( \lim_{x \to 2} \frac{x^2 – 4}{x – 2} \approx 4 \).

Example :

Use the table to estimate \( \lim_{x \to 0} \frac{\sin x}{x} \) (in radians).

\( x \)\( f(x) = \frac{\sin x}{x} \)
-0.10.9983
-0.010.99998
-0.0010.9999998
0.0010.9999998
0.010.99998
0.10.9983
▶️ Answer/Explanation

The function values approach 1 as \( x \to 0 \) from both sides.

\( \lim_{x \to 0} \frac{\sin x}{x} \approx 1 \).

Example :

The function \( f(x) \) has the following values near \( x = 1 \).

Use the table to estimate \( \lim_{x \to 1^-} f(x) \), \( \lim_{x \to 1^+} f(x) \), and determine if \( \lim_{x \to 1} f(x) \) exists.

\( x \)\( f(x) \)
0.94.9
0.994.99
0.9994.999
1.0018.001
1.018.01
1.18.1
▶️ Answer/Explanation

From the left: As \( x \to 1^- \), the values approach 5, so \( \lim_{x \to 1^-} f(x) = 5 \).

From the right: As \( x \to 1^+ \), the values approach 8, so \( \lim_{x \to 1^+} f(x) = 8 \).

Since LHL ≠ RHL, the two-sided limit does not exist: \( \lim_{x \to 1} f(x) \text{ does not exist} \).

More Workout Examples

Exercises – Properties of Continuity and Intermediate Value Theorem
Multiple Choice Questions

Question

  •  Let f be a function defined by \(f\left ( x \right )=\left\{\begin{matrix}\frac{x^{2}-a^{2}}{x-a}, & if x\neq a\\4 & if x = a\end{matrix}\right.\). If f is continuous for all real numbers x , what is the value of a ?

(A) \(\frac{1}{2}\)                (B) 0                (C) 1                (D) 2

▶️Answer/Explanation

Ans:

1. D

Question

  •  The graph of a function f is shown above. If \(\underset{x\rightarrow a}{lim}f\left ( x \right )\) exists and f is not continuous at x = a, then a =

(A) -1                (B) 0                (C) 2                (D) 4

▶️Answer/Explanation

Ans:

2. C

Question

  • If \(f\left ( x \right )=\left\{\begin{matrix}\frac{\sqrt{3x-1}-\sqrt{2x}}{x-1}, & for x\neq 1\\a, & for x = 1\end{matrix}\right.,\) and if f is continuous at x = 1 , then a =

(A) \(\frac{1}{4}\)                (B) \(\frac{\sqrt{2}}{4}\)                (C) \(\sqrt{2}\)                (D) 2

▶️Answer/Explanation

Ans:

3. B

Question

  • Let f be a continuous function on the closed interval [−2,7]. If f (-2) = 5 and f (7) = -3, then the Intermediate Value Theorem guarantees that

(A) f′(c) = 0 for at least one c between −2 and 7

(B) f′(c) = 0 for at least one c between −3 and 5

(C) f′(c) = 0 for at least one c between −3 and 5

(D) f′(c) = 0 for at least one c between −2 and 7

▶️Answer/Explanation

Ans:

4. D

Free Response Questions

Question

  •  Let g be a function defined by \(g\left ( x \right )=\left\{\begin{matrix}\frac{\pi sin x}{x}, & if x< 0\\a-bx, & if 0\leq x< 1.\\arctan x, & if x\geq 1\end{matrix}\right.\)

     If g is continuous for all real numbers x , what are the values of a and b ?

▶️Answer/Explanation

Ans:

5. \(a=\pi , b=\frac{3\pi }{4}\)

Question

  •  Evaluate \(\underset{a\rightarrow 0}{lim}\frac{-1+\sqrt{1+a}}{a}\).
▶️Answer/Explanation

Ans:

6. \(\frac{1}{2}\)

Question

  • What is the value of a , if \(\underset{x\rightarrow 0}{lim}\frac{\sqrt{ax+9-3}}{x}=1\)?
▶️Answer/Explanation

Ans:

7. 6

Scroll to Top