Question
![1 fourth](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABIAAAAjCAYAAACZ6FpfAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAXQ/cXWQAAAGxJREFUeNpjYMAP1IC4FogvMFAIFgNxGhD/Z6ASGDVo1KBBYdB/LHgUDCbwn0w8CgYr8KFGDPEA8TVqGDQTiJMpNcgaiPdSWiaxAfElIJan1KAOIM6htJTUA+Kj1ChuTwKxCjUMommOH2J1GwCVNzk6FQ24cQAAAGJ0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bWZyYWM+PG1uPjE8L21uPjxtbj40PC9tbj48L21mcmFjPjwvbWF0aD7CA6ZrAAAAAElFTkSuQmCC)
![1 half](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABIAAAAjCAYAAACZ6FpfAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAXQ/cXWQAAAJFJREFUeNpjYMAP1IC4FogvMFAIFgNxGhD/Z6ASGDVo1KBBYdB/LHgUDCbwn0w8CgYLsAbiNUD8CYh/QaujaHIMOgjEkUDMA+VrAfFRqBjFQB6IL1HLyz+oYYgl1HsUAQ4gPgmNBLKBIBBvAGI3SgxRghqiQokhGkA8G4i5KDFEHIhXATELpYG7BeoimhZyWAEAI3M1I31CbrEAAABidEVYdE1hdGhNTAA8bWF0aCB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMTk5OC9NYXRoL01hdGhNTCI+PG1mcmFjPjxtbj4xPC9tbj48bW4+MjwvbW4+PC9tZnJhYz48L21hdGg+ND6jrQAAAABJRU5ErkJggg==)
- 2
- 4
Hint:
We can apply L'Hopital's rule, also commonly spelled L'Hospital's rule, whenever direct substitution of a limit yields an indeterminate form. This means that the limit of a quotient of functions (i.e., an algebraic fraction) is equal to the limit of their derivatives.
In this question, we have to find value of
.
The correct answer is: ![1 fourth](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABIAAAAjCAYAAACZ6FpfAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAXQ/cXWQAAAGxJREFUeNpjYMAP1IC4FogvMFAIFgNxGhD/Z6ASGDVo1KBBYdB/LHgUDCbwn0w8CgYr8KFGDPEA8TVqGDQTiJMpNcgaiPdSWiaxAfElIJan1KAOIM6htJTUA+Kj1ChuTwKxCjUMommOH2J1GwCVNzk6FQ24cQAAAGJ0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bWZyYWM+PG1uPjE8L21uPjxtbj40PC9tbj48L21mcmFjPjwvbWF0aD7CA6ZrAAAAAElFTkSuQmCC)
![Lt subscript x not stretchy rightwards arrow 4 end subscript space fraction numerator square root of x minus 2 over denominator x minus 4 end fraction](data:image/png;base64,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)
We first try substitution:
=
=
( L'Hopital's Rule for zero over zero.)
Since the limit is in the form
, it is indeterminate we don’t yet know what is it. We need to do some work to put it in a form where we can determine the limit.
(
)
We can write simply,
=
=![1 fourth](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABIAAAAjCAYAAACZ6FpfAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAXQ/cXWQAAAGxJREFUeNpjYMAP1IC4FogvMFAIFgNxGhD/Z6ASGDVo1KBBYdB/LHgUDCbwn0w8CgYr8KFGDPEA8TVqGDQTiJMpNcgaiPdSWiaxAfElIJan1KAOIM6htJTUA+Kj1ChuTwKxCjUMommOH2J1GwCVNzk6FQ24cQAAAGJ0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bWZyYWM+PG1uPjE8L21uPjxtbj40PC9tbj48L21mcmFjPjwvbWF0aD7CA6ZrAAAAAElFTkSuQmCC)
We can only apply the L’Hospital’s rule if the direct substitution returns an indeterminate form, that means or
.
Related Questions to study
Hence Choice 4 is correct
Hence Choice 4 is correct
We can only apply the L’Hospital’s rule if the direct substitution returns an indeterminate form, that means or
.
We can only apply the L’Hospital’s rule if the direct substitution returns an indeterminate form, that means or
.
We can only apply the L’Hospital’s rule if the direct substitution returns an indeterminate form, that means .
We can only apply the L’Hospital’s rule if the direct substitution returns an indeterminate form, that means .