Question:
\(
\lim_{x \to 3} \frac{x^3 - 27}{x^2 - 9}.
\)
\(
\lim_{x \to 3} \frac{x^3 - 27}{x^2 - 9}.
\)
Show Hint
When evaluating a limit in the form \( \frac{0}{0} \), first try factorization or L'Hôpital's Rule to simplify the expression.
Updated On: Mar 19, 2025
- \( \frac{3}{2} \)
- \( \frac{9}{2} \)
- \( 3 \)
- \( 2 \)
Hide Solution
Verified By Collegedunia
The Correct Option is B
Solution and Explanation
Step 1: Factorizing the numerator and denominator \( x^3 - 27 = (x-3)(x^2 + 3x + 9). \) \( x^2 - 9 = (x-3)(x+3). \)
Step 2: Cancel common terms \( \frac{(x-3)(x^2+3x+9)}{(x-3)(x+3)}. \) For \( x \neq 3 \), canceling \( (x-3) \), \( \lim_{x \to 3} \frac{x^2+3x+9}{x+3}. \)
Step 3: Substitute \( x = 3 \) \( \frac{3^2+3(3)+9}{3+3} = \frac{9+9+9}{6} = \frac{27}{6} = \frac{9}{2}. \)
Was this answer helpful?
0
0
Top Questions on Limits
- If \( f(x) = \begin{cases} \frac{(e^x - 1) \log(1 + x)}{x^2} & \text{if } x>0 \\ 1 & \text{if } x = 0 \\ \frac{\cos 4x - \cos bx}{\tan^2 x} & \text{if } x<0 \end{cases} \) is continuous at \( x = 0 \), then \(\sqrt{b^2 - a^2} =\)
- \(\lim_{x \to 0} \frac{x \tan 2x - 2x \tan x}{(1 - \cos 2x)^2} =\)
- Let \([x]\) represent the greatest integer function. If \(\lim_{x \to 0^+} \frac{\cos[x] - \cos(kx - [x])}{x^2} = 5\), then \(k =\)
- Evaluate the limit: \[ \lim_{x \to 0} \frac{(\csc x - \cot x)(e^x - e^{-x})}{\sqrt{3} - \sqrt{2 + \cos x}} \]
- If a real valued function \( f(x) = \begin{cases} (1 + \sin x)^{\csc x} & , -\frac{\pi}{2} < x < 0 \\ a & , x = 0 \\ \frac{e^{2/x} + e^{3/x}}{ae^{2/x} + be^{3/x}} & , 0 < x < \frac{\pi}{2} \end{cases} \) is continuous at \(x = 0\), then \(ab = \)
View More Questions
Questions Asked in AP EAMCET exam
- A particle of mass 2 g and charge 6 $\mu$C is accelerated from rest through a potential difference of 60 V. The speed acquired by the particle is:
- AP EAMCET - 2024
- Speed and velocity
- P, Q, and R try to hit the same target one after another. Their probabilities of hitting are \( \frac{2}{3}, \frac{3}{5}, \frac{5}{7} \) respectively. Find the probability that the target is hit by P or Q but not by R.
- AP EAMCET - 2024
- Probability
- In a meter bridge experiment, a resistance of 9 Ω is connected in the left gap and an unknown resistance greater than 9 Ω is connected in the right gap. If the resistance in the gaps are interchanged, the balancing point shifts by 10 cm. The unknown resistance is:
- AP EAMCET - 2024
- Wheatstone Bridge
- The angle made by the resultant vector of two vectors \( 2\hat{i} + 3\hat{j} + 4\hat{k} \) and \( 2\hat{i} - 7\hat{j} - 4\hat{k} \) with the x-axis is:
- If the longest wavelength of the spectral line of the Paschen series of \( \text{Li}^{2+} \) ion spectrum is \( x \) Å, then the longest wavelength (in Å) of the Lyman series of the hydrogen spectrum is:
- AP EAMCET - 2024
- Atomic Spectra
View More Questions