LIMITS AND DERIVATIVES — MATHS STD 11 Science — Question
Gujarat BoardEnglish MediumSTD 11 ScienceMATHSLIMITS AND DERIVATIVES1 Mark
Question
Evaluate the following limits in Exercise: $\lim\limits_{\text{x} \rightarrow-0}\frac{(\text{x}+1)^5-1}{\text{x}}$
✓
Answer
$\lim\limits_{\text{x} \rightarrow-0}\frac{(\text{x}+1)^5-1}{\text{x}}$Put x + 1 = y so that y → 1 as x → 0.Put x + 1 = y so that y → 1 as x → 0.
$\text{Accordingly},\lim\limits_{\text{x}\rightarrow0}\frac{(\text{x}+1)^5-1}{\text{x}}=\lim\limits_{\text{y}\rightarrow1}\frac{\text{y}^5-1}{\text{y}-1}$
$=5.1^{5-1}$ $\Big[\lim\limits_{\text{x}\rightarrow\text{a}}\frac{\text{x}^\text{n}-\text{a}^\text{n}}{\text{x}-\text{a}}=\text{na}^{\text{n}-1}\Big]$
$=5$
$\therefore\lim\limits_{\text{x}\rightarrow0}\frac{(\text{x}+5)^5-1}{\text{x}}=5$
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