Question
- 1
- 0
- 2
![1 half](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABIAAAAjCAYAAACZ6FpfAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAXQ/cXWQAAAJFJREFUeNpjYMAP1IC4FogvMFAIFgNxGhD/Z6ASGDVo1KBBYdB/LHgUDCbwn0w8CgYLsAbiNUD8CYh/QaujaHIMOgjEkUDMA+VrAfFRqBjFQB6IL1HLyz+oYYgl1HsUAQ4gPgmNBLKBIBBvAGI3SgxRghqiQokhGkA8G4i5KDFEHIhXATELpYG7BeoimhZyWAEAI3M1I31CbrEAAABidEVYdE1hdGhNTAA8bWF0aCB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMTk5OC9NYXRoL01hdGhNTCI+PG1mcmFjPjxtbj4xPC9tbj48bW4+MjwvbW4+PC9tZnJhYz48L21hdGg+ND6jrQAAAABJRU5ErkJggg==)
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: 2
![Lt subscript x not stretchy rightwards arrow 0 end subscript space fraction numerator e to the power of x minus 1 over denominator square root of 1 plus x end root minus 1 end fraction](data:image/png;base64,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)
We first try substitution :
=
=
= ![0 over 0](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABIAAAAjCAYAAACZ6FpfAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAdoyL+RwAAAJtJREFUeNpjYMAN+IB4GhCfhOKZUDGSwV4g9kPi+0DFSAKhQDwJi/gEII4kxaA1QOyGRdwRKkc0eAfEbFjEQWIvSTHoDx65X9Qy6AcpBr3E4TUOUr0GClAPLOJupAZ2EBDPxiI+n9ToB4H9QFwAxCxQb1YD8UFyUrYgEM+FBu43aBbhYRgFgwv8JxOPgsEERquj0epotDqiXnUEAHl3QY/fxbExAAAAf3RFWHRNYXRoTUwAPG1hdGggeG1sbnM9Imh0dHA6Ly93d3cudzMub3JnLzE5OTgvTWF0aC9NYXRoTUwiPjxtZnJhYz48bW4+MDwvbW4+PG1uPjA8L21uPjwvbWZyYWM+PG1zcGFjZSBsaW5lYnJlYWs9Im5ld2xpbmUiLz48L21hdGg+lAEH4gAAAABJRU5ErkJggg==)
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 know,
)
![Lt subscript x not stretchy rightwards arrow 0 end subscript space fraction numerator e to the power of x minus 1 over denominator x end fraction cross times space Lt subscript x not stretchy rightwards arrow 0 end subscript space fraction numerator x over denominator square root of 1 plus x end root minus 1 end fraction](data:image/png;base64,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)
=
( Let
= y, we know
)
Now, We can write simply
=
=2
We can only apply the L’Hospital’s rule if the direct substitution returns an indeterminate form, that means .
Related Questions to study
If
then
If
then
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
.
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 .