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Evaluate the limit: lim (x->0) (5x) / (x2 + 2x)

Mathematics

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Answer

52\frac{5}{2}

Step-by-step solution

4 steps

  1. Substitute x=0x=0 into the expression to check for an indeterminate form.

    5(0)02+2(0)=00\frac{5(0)}{0^2 + 2(0)} = \frac{0}{0}

    Since this is an indeterminate form, we need to simplify the expression.

  2. Factor out xx from the denominator.

    lim⁡x→05xx(x+2)\lim_{x \to 0} \frac{5x}{x(x+2)}

  3. Cancel out the common factor xx from the numerator and denominator.

    This is valid because x→0x \to 0 means xx is approaching 0 but is not equal to 0.

    lim⁡x→05x+2\lim_{x \to 0} \frac{5}{x+2}

  4. Substitute x=0x=0 into the simplified expression.

    50+2=52\frac{5}{0+2} = \frac{5}{2}

    The final answer is 52\frac{5}{2}.

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