Question
- 1
- 0
- -1
![e squared](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABIAAAARCAYAAADQWvz5AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAQ3ZOC+gAAAKVJREFUeNpjYCAN/IfiX0B8FIhVGCgETECcBcTnGKgEflDDEHsgPkipIZLQMFKixBA1IN4CNYwil2wDYj5CCsWBeA00EB9CwwEZgAzRIGQIyJZLQOwH5WsB8RUc6QgZY4AuIK5EEwOFBRupCewDEIuiiX2C0kQDcxzOPkpqbIDCZRU1UqklEJ+kVr65BsQF0DARhAa8LTkGgZL7YSD+AzU0mWEgAAAoyiWYZ6Ex8AAAAGB0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bXN1cD48bWk+ZTwvbWk+PG1uPjI8L21uPjwvbXN1cD48L21hdGg+JeSUYAAAAABJRU5ErkJggg==)
Hint:
In this question, we have to find value of ![L t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator open parentheses 1 minus e to the power of x close parentheses sin begin display style space end style x over denominator x squared plus x cubed end fraction.](data:image/png;base64,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)
The correct answer is: -1
![L t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator open parentheses 1 minus e to the power of x close parentheses sin begin display style space end style x over denominator x squared plus x cubed end fraction](data:image/png;base64,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)
![L t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator open parentheses 1 minus e to the power of x close parentheses sin begin display style space end style x over denominator x squared plus x cubed end fraction space equals space fraction numerator open parentheses 1 minus e to the power of 0 close parentheses sin begin display style space end style 0 over denominator 0 end fraction space equals space 0 over 0](data:image/png;base64,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)
Since the limit is in the form 0 over 0, 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.
![L t subscript straight x not stretchy rightwards arrow 0 end subscript fraction numerator stretchy left parenthesis 1 minus straight e to the power of straight x stretchy right parenthesis sin begin display style space end style straight x over denominator straight x left parenthesis 1 plus straight x right parenthesis straight x end fraction space space space left parenthesis W e space k n o w space t h a t space space L t subscript straight x not stretchy rightwards arrow 0 end subscript fraction numerator sin begin display style space end style straight x over denominator straight x end fraction equals 1 comma space L t subscript straight x not stretchy rightwards arrow 0 end subscript fraction numerator left parenthesis straight e to the power of straight x minus 1 right parenthesis over denominator straight x end fraction equals 1 right parenthesis
L t subscript straight x not stretchy rightwards arrow 0 end subscript fraction numerator negative stretchy left parenthesis straight e to the power of straight x minus 1 stretchy right parenthesis over denominator straight x left parenthesis 1 plus straight x right parenthesis end fraction cross times fraction numerator sinx space over denominator straight x end fraction
L t subscript straight x not stretchy rightwards arrow 0 end subscript fraction numerator negative 1 over denominator left parenthesis 1 plus straight x right parenthesis end fraction
p u t space x equals 0 space t h e n comma
fraction numerator negative 1 over denominator left parenthesis 1 plus 0 right parenthesis end fraction equals space minus 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Related Questions to study
We can only apply the L’Hospital’s rule if the direct substitution returns an indeterminate form, that means 0 over 0 or
We can only apply the L’Hospital’s rule if the direct substitution returns an indeterminate form, that means 0 over 0 or
We can only apply the L’Hospital’s rule if the direct substitution returns an indeterminate form, that means 0 over 0 or
We can only apply the L’Hospital’s rule if the direct substitution returns an indeterminate form, that means 0 over 0 or
The correct relation is.
The correct relation is.
A vessel whose bottom has round holes with diameter of 0.1 mm is filled with water. The maximum height to which the water can be filled without leakage is
(S.T. of water )
A vessel whose bottom has round holes with diameter of 0.1 mm is filled with water. The maximum height to which the water can be filled without leakage is
(S.T. of water )
We can only apply the L’Hospital’s rule if the direct substitution returns an indeterminate form, that means 0 over 0 or
We can only apply the L’Hospital’s rule if the direct substitution returns an indeterminate form, that means 0 over 0 or
A capillary tube at radius R is immersed in water and water rises in it to a height H. Mass of water in the capillary tube is M. If the radius of the tube is doubled. Mass of water that will rise in the, capillary tube will now be
A capillary tube at radius R is immersed in water and water rises in it to a height H. Mass of water in the capillary tube is M. If the radius of the tube is doubled. Mass of water that will rise in the, capillary tube will now be
Direct substitution can sometimes be used to calculate the limits for functions involving trigonometric functions.
Direct substitution can sometimes be used to calculate the limits for functions involving trigonometric functions.
If the excess pressure inside a soap bubble is balanced by oil column of height 2 mm then the surface tension of soap solution will be
If the excess pressure inside a soap bubble is balanced by oil column of height 2 mm then the surface tension of soap solution will be
The quadratic equation whose roots are I and m where
is
The quadratic equation whose roots are I and m where
is
Two bubbles A and B (A>B) are joined through a narrow tube than,
Two bubbles A and B (A>B) are joined through a narrow tube than,
Direct substitution can sometimes be used to calculate the limits for functions involving trigonometric functions.
Direct substitution can sometimes be used to calculate the limits for functions involving trigonometric functions.
In capillary pressure below the curved surface at water will be
In capillary pressure below the curved surface at water will be
is equal to
The substitution rule for calculating limits is a method of finding limits, by simply substituting the value of x with the point at which we want to calculate the limit.
is equal to
The substitution rule for calculating limits is a method of finding limits, by simply substituting the value of x with the point at which we want to calculate the limit.