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General
Easy

Question

L t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator e to the power of x minus e to the power of in space x end exponent over denominator 2 left parenthesis x minus sin space x right parenthesis end fraction equals

  1. 1
  2. 2
  3. -1
  4. 1 half

The correct answer is: 1 half

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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 equals

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 equals

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text  If  end text a greater than 0 text  and end text L subscript x not stretchy rightwards arrow a end subscript fraction numerator a to the power of a minus x to the power of a over denominator x to the power of a minus a to the power of a end fraction equals negative 1 text , then  end text bold a equals

We can only apply the L’Hospital’s rule if the direct substitution returns an indeterminate form, that means 0 over 0 or  fraction numerator plus-or-minus infinity over denominator plus-or-minus infinity end fraction.

text  If  end text a greater than 0 text  and end text L subscript x not stretchy rightwards arrow a end subscript fraction numerator a to the power of a minus x to the power of a over denominator x to the power of a minus a to the power of a end fraction equals negative 1 text , then  end text bold a equals

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We can only apply the L’Hospital’s rule if the direct substitution returns an indeterminate form, that means 0 over 0 or  fraction numerator plus-or-minus infinity over denominator plus-or-minus infinity end fraction.

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L subscript x not stretchy rightwards arrow 0 end subscript fraction numerator x left parenthesis 1 plus a cos space x right parenthesis minus b sin space x over denominator x cubed end fraction equals 1 text  then  end text straight a equals comma straight b equals

We can only apply the L’Hospital’s rule if the direct substitution returns an indeterminate form, that means 0 over 0 or fraction numerator plus-or-minus infinity over denominator plus-or-minus infinity end fraction.

L subscript x not stretchy rightwards arrow 0 end subscript fraction numerator x left parenthesis 1 plus a cos space x right parenthesis minus b sin space x over denominator x cubed end fraction equals 1 text  then  end text straight a equals comma straight b equals

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We can only apply the L’Hospital’s rule if the direct substitution returns an indeterminate form, that means 0 over 0 or fraction numerator plus-or-minus infinity over denominator plus-or-minus infinity end fraction.

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The correct relation is.

The correct relation is.

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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 == fraction numerator 75 text  dyne  end text over denominator cm end fraction straight g equals 1000 straight m over straight s squared times)

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 == fraction numerator 75 text  dyne  end text over denominator cm end fraction straight g equals 1000 straight m over straight s squared times)

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L t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator x 10 to the power of x minus x over denominator 1 minus cos space x end fraction equals

We can only apply the L’Hospital’s rule if the direct substitution returns an indeterminate form, that means 0 over 0 or fraction numerator plus-or-minus infinity over denominator plus-or-minus infinity end fraction.

L t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator x 10 to the power of x minus x over denominator 1 minus cos space x end fraction equals

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We can only apply the L’Hospital’s rule if the direct substitution returns an indeterminate form, that means 0 over 0 or fraction numerator plus-or-minus infinity over denominator plus-or-minus infinity end fraction.

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L t subscript n not stretchy rightwards arrow straight infinity end subscript fraction numerator n open parentheses 1 cubed plus 2 cubed plus midline horizontal ellipsis times plus n cubed close parentheses squared over denominator open parentheses 1 squared plus 2 squared plus midline horizontal ellipsis plus n squared close parentheses cubed end fraction equals

In general, we say that f(x) tends to a real limit l as x tends to infinity if, however small a distance we choose, f(x) gets closer than that distance to l and stays closer as x increases. f(x) = infinity

L t subscript n not stretchy rightwards arrow straight infinity end subscript fraction numerator n open parentheses 1 cubed plus 2 cubed plus midline horizontal ellipsis times plus n cubed close parentheses squared over denominator open parentheses 1 squared plus 2 squared plus midline horizontal ellipsis plus n squared close parentheses cubed end fraction equals

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In general, we say that f(x) tends to a real limit l as x tends to infinity if, however small a distance we choose, f(x) gets closer than that distance to l and stays closer as x increases. f(x) = infinity

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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

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Lt subscript x not stretchy rightwards arrow 0 end subscript fraction numerator 1 over denominator sin squared space x end fraction minus fraction numerator 1 over denominator space x squared end fraction

Direct substitution can sometimes be used to calculate the limits for functions involving trigonometric functions.

Lt subscript x not stretchy rightwards arrow 0 end subscript fraction numerator 1 over denominator sin squared space x end fraction minus fraction numerator 1 over denominator space x squared end fraction

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Direct substitution can sometimes be used to calculate the limits for functions involving trigonometric functions.

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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 C r equals 1 space a n d space d e n s i t y space d equals 0.8 g m divided by c c right parenthesis

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 C r equals 1 space a n d space d e n s i t y space d equals 0.8 g m divided by c c right parenthesis

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The quadratic equation whose roots are I and m wherel equals lim for theta not stretchy rightwards arrow 0 of   open parentheses fraction numerator 3 sin space theta minus 4 sin cubed space theta over denominator theta end fraction close parentheses m equals lim for theta not stretchy rightwards arrow 0 of   open parentheses fraction numerator 2 tan space theta over denominator theta open parentheses 1 minus tan squared space theta close parentheses end fraction close parentheses is

The quadratic equation whose roots are I and m wherel equals lim for theta not stretchy rightwards arrow 0 of   open parentheses fraction numerator 3 sin space theta minus 4 sin cubed space theta over denominator theta end fraction close parentheses m equals lim for theta not stretchy rightwards arrow 0 of   open parentheses fraction numerator 2 tan space theta over denominator theta open parentheses 1 minus tan squared space theta close parentheses end fraction close parentheses is

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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,

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L subscript x not stretchy rightwards arrow 0 end subscript fraction numerator tan cubed begin display style space end style x minus sin cubed space x over denominator x cubed end fraction

Direct substitution can sometimes be used to calculate the limits for functions involving trigonometric functions.

L subscript x not stretchy rightwards arrow 0 end subscript fraction numerator tan cubed begin display style space end style x minus sin cubed space x over denominator x cubed end fraction

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Direct substitution can sometimes be used to calculate the limits for functions involving trigonometric functions.

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In capillary pressure below the curved surface at water will be

In capillary pressure below the curved surface at water will be

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L subscript x not stretchy rightwards arrow 0 end subscript fraction numerator sin begin display style space end style open parentheses pi cos squared space x close parentheses over denominator x squared end fraction 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.

L subscript x not stretchy rightwards arrow 0 end subscript fraction numerator sin begin display style space end style open parentheses pi cos squared space x close parentheses over denominator x squared end fraction is equal to

Maths-General

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.

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