Advertisements
Advertisements
प्रश्न
Find \[\lim_{x \to 3^+} \frac{x}{\left[ x \right]} .\] Is it equal to \[\lim_{x \to 3^-} \frac{x}{\left[ x \right]} .\]
उत्तर
\[\lim_{x \to 3^+} \frac{x}{\left[ x \right]}\]
\[\text{ Let } x = 3 + \text{ h, where h } \to 0 . \]
\[ \Rightarrow \lim_{h \to 0} \left( \frac{3 + h}{\left[ 3 + h \right]} \right) = \frac{3}{3} = 1\]
\[Also, \lim_{x \to 3^-} \left( \frac{x}{\left[ x \right]} \right)\]
\[\text{ Let } x = 3 - \text{ h, where h } \to 0 . \]
\[ \lim_{h \to 0} \left( \frac{3 - h}{\left[ 3 - h \right]} \right)\]
\[ = \frac{3}{2}\]
\[ \therefore \lim_{x \to 3^-} \frac{x}{\left[ x \right]} \neq \lim_{x \to 3^+} \frac{x}{\left[ x \right]}\]
APPEARS IN
संबंधित प्रश्न
Find k so that \[\lim_{x \to 2} f\left( x \right)\] \[f\left( x \right) = \begin{cases}2x + 3, & x \leq 2 \\ x + k, & x > 2\end{cases} .\]
Show that \[\lim_{x \to 0} \frac{1}{x}\] does not exist.
Let f(x) be a function defined by \[f\left( x \right) = \begin{cases}\frac{3x}{\left| x \right| + 2x}, & x \neq 0 \\ 0, & x = 0\end{cases} .\] Show that \[\lim_{x \to 0} f\left( x \right)\] does not exist.
Let \[f\left( x \right) = \left\{ \begin{array}{l}x + 1, & if x \geq 0 \\ x - 1, & if x < 0\end{array} . \right.\]Prove that \[\lim_{x \to 0} f\left( x \right)\] does not exist.
Find \[\lim_{x \to 3} f\left( x \right)\] where \[f\left( x \right) = \begin{cases}4, & if x > 3 \\ x + 1, & if x < 3\end{cases}\]
If \[f\left( x \right) = \left\{ \begin{array}{l}2x + 3, & x \leq 0 \\ 3 \left( x + 1 \right), & x > 0\end{array} . \right.\] find \[\lim_{x \to 0} f\left( x \right)\]
If \[f\left( x \right) = \left\{ \begin{array}{l}2x + 3, & x \leq 0 \\ 3 \left( x + 1 \right), & x > 0\end{array} . \right.\] find \[\lim_{x \to 1} f\left( x \right)\]
Find \[\lim_{x \to 1} f\left( x \right)\] if \[f\left( x \right) = \begin{cases}x^2 - 1, & x \leq 1 \\ - x^2 - 1, & x > 1\end{cases}\]
Evaluate \[\lim_{x \to 0} f\left( x \right)\] where \[f\left( x \right) = \begin{cases}\frac{\left| x \right|}{x}, & x \neq 0 \\ 0, & x = 0\end{cases}\]
Find \[\lim_{x \to 1^+} \left( \frac{1}{x - 1} \right) .\]
Evaluate the following one sided limit:
\[\lim_{x \to 0^+} \frac{1}{3x}\]
Evaluate the following one sided limit:
\[\lim_{x \to - 8^+} \frac{2x}{x + 8}\]
Evaluate the following one sided limit:
\[\lim_{x \to \frac{\pi}{2}} \tan x\]
Evaluate the following one sided limit:
\[\lim_{x \to 0^-} 2 - \cot x\]
Evaluate the following one sided limit:
\[\lim_{x \to 0^-} 1 + cosec x\]
Show that \[\lim_{x \to 0} e^{- 1/x}\] does not exist.
Find: \[\ \lim_{x \to 2} \left[ x \right]\]
Find: \[ \lim_{x \to \frac{5}{2}} \left[ x \right]\]
Show that \[\lim_{x \to 2^-} \frac{x}{\left[ x \right]} \neq \lim_{x \to 2^+} \frac{x}{\left[ x \right]} .\]
Find \[\lim_{x \to 5/2} \left[ x \right] .\]
Evaluate \[\lim_{x \to 2} f\left( x \right)\] (if it exists), where \[f\left( x \right) = \left\{ \begin{array}{l}x - \left[ x \right], & x < 2 \\ 4, & x = 2 \\ 3x - 5, & x > 2\end{array} . \right.\]
Show that \[\lim_{x \to 0} \sin \frac{1}{x}\]does not exist.
Let \[f\left( x \right) = \begin{cases}\frac{k\cos x}{\pi - 2x}, & where x \neq \frac{\pi}{2} \\ 3, & where x = \frac{\pi}{2}\end{cases}\] and if \[\lim_{x \to \frac{\pi}{2}} f\left( x \right) = f\left( \frac{\pi}{2} \right)\]
