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
- The function \[ f(x)=\sin 2x+2\cos x,\qquad x\in\left(-\frac{3\pi}{4},\frac{3\pi}{4}\right) \] has:
- Let \(f\) be a differentiable function satisfying \[ f(x+y)=f(x)+f(y)-xy \quad \text{for all } x,y\in\mathbb{R}. \] If \[ \lim_{h\to 0}\frac{f(h)}{h}=3, \] then the value of \[ \sum_{n=1}^{10} f(n) \] is equal to:
- Let \(f:[-2a,2a]\to\mathbb{R}\) be a thrice differentiable function and define \[ g(x)=f(a+x)+f(a-x). \] If \(m\) is the minimum number of roots of \(g'(x)=0\) in the interval \((-a,a)\) and \(n\) is the minimum number of roots of \(g''(x)=0\) in the interval \((-a,a)\), then \(m+n\) is equal to:
- If \( f(3) = 18, f'(3) = 0 \) and \( f''(3) = 4 \), then the value of \[ \lim_{x \to 3} \ln \left( \frac{f(x+2)}{f(3)} \right)^{\frac{18}{(x-3)^3}} \] is equal to
- Evaluate the limit: \[ \lim_{x \to 0} \frac{e^{x}\left(e^{\tan x - x} - 1\right) + \ell n(\sec x + \tan x) - x}{\tan x - x} \]
View More Questions
Questions Asked in AP EAMCET exam
Which of the following are ambident nucleophiles?
[A.] CN$^{\,-}$
[B.] CH$_{3}$COO$^{\,-}$
[C.] NO$_{2}^{\,-}$
[D.] CH$_{3}$O$^{\,-}$
[E.] NH$_{3}$- AP EAMCET - 2024
- Acids and Bases
- The correct sequence of enzymes involved in the commercial production of ethanol by fermentation from sugar is:
- AP EAMCET - 2024
- Acids and Bases
- At 298 K, the ionization constant of \( {CN}^- \) is \( 2.08 \times 10^{-6} \). What is the ionization constant of its conjugate acid? (Given \( K_w = 10^{-14} \))
- AP EAMCET - 2024
- Acids and Bases
Identify the anomers from the following.
- AP EAMCET - 2024
- Acids and Bases
The standard Gibbs free energy change \( \Delta G^\circ \) of a cell reaction is \(-301 { kJ/mol}\). What is \( E^\circ \) in volts?
(Given: \( F = 96500 { C/mol}\), \( n = 2 \))- AP EAMCET - 2024
- Acids and Bases
View More Questions