How do solve without using derivative?

#lim xrarr0 (sin(x^3+x^2-x)+sinx)/x#

Answer 1

#lim_(x->0)frac (sin(x^3+x^2-x)+sinx) x =0#

You can resolve this limit without using derivatives by re-conducing it to the known limit:

#lim_(x->0)sinx/x=1#

We can proceed in this way:

#frac (sin(x^3+x^2-x)+sinx) x = frac (sin(x^3+x^2-x) )x + frac (sinx) x = frac (sin(x^3+x^2-x) ) (x^3+x^2-x)* frac (x^3+x^2-x) x+ frac (sinx) x = frac (sin(x^3+x^2-x) ) (x^3+x^2-x)* (x^2+x-1) + frac (sinx) x #

Thus: #lim_(x->0)frac (sin(x^3+x^2-x)+sinx) x =lim_(x->0)frac (sin(x^3+x^2-x) ) (x^3+x^2-x)* (x^2+x-1) + frac (sinx) x = 1*(-1)+1=0#

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Answer 2

Scarlett Allison

To solve a problem without using derivatives, you can use alternative methods such as algebraic manipulation, graphical analysis, or numerical approximation techniques. The specific approach will depend on the problem you are trying to solve. Can you please provide more details or specify the problem you need assistance with?

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.