Evaluate the following limit : lim(x→a) (x^2/3 - a^2/3)/(x^3/4 - a^3/4)

Question

Evaluate the following limit : \(\lim\limits_{\text x \to a}\cfrac{\text x^{2/3}-a^{2/3}}{\text x^{3/4}-a^{3/4}}\)

lim(x→a) (x^2/3 - a^2/3)/(x^3/4 - a^3/4)

Answer 1

We need to find the limit for: \(\lim\limits_{\text x \to a}\cfrac{\text x^{2/3}-a^{2/3}}{\text x^{3/4}-a^{3/4}}\)

As limit can’t be find out simply by putting x = a because it is taking indeterminate form(0/0) form, so we need to have a different approach.

Let, Z = \(\lim\limits_{\text x \to a}\cfrac{\text x^{2/3}-a^{2/3}}{\text x^{3/4}-a^{3/4}}\)

 Note: To solve the problems of limit similar to one in our question we use the formula mentioned below which can be derived using binomial theorem.

Formula to be used: \(\lim\limits_{\text x \to a}\cfrac{(\text x)^n-(a)^n}{\text x-a} \) = na^{n -1}

As Z does not match exactly with the form as described above so we need to do some manipulations–

Dividing numerator and denominator by (x – a),we get

Using algebra of limits, we have –

Use the formula: \(\lim\limits_{\text x \to a}\cfrac{(\text x)^n-(a)^n}{\text x-a} \) = na^{n -1}

Hence, \(\lim\limits_{\text x \to a}\cfrac{\text x^{2/3}-a^{2/3}}{\text x^{3/4}-a^{3/4}}\)