Immediate limits and indeterminate forms

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Compute the following limits:

• (a) $\displaystyle\lim_{x\to2}(x^{2}+3x-1)$
• (b) $\displaystyle\lim_{x\to+\infty}\frac{3x^{2}-x}{x^{2}+5}$
• (c) $\displaystyle\lim_{x\to0}\frac{\sin 4x}{x}$
• (d) $\displaystyle\lim_{x\to0}\frac{\sqrt{x+1}-1}{x}$

Solution

(a) Direct substitution: $4+6-1=9$.

(b) Polynomials of equal degree: limit $=\dfrac{3}{1}=3$.

(c) $\dfrac{\sin4x}{x}=4\cdot\dfrac{\sin4x}{4x}\to4$ (standard limit).

(d) Rationalizing: $\dfrac{\sqrt{x+1}-1}{x}=\dfrac{1}{\sqrt{x+1}+1}\to\dfrac12$.

$\boxed{a=9;\ b=3;\ c=4;\ d=\tfrac12}$