2. Function Limits

Function Limits

Ex 1.

Calculate the function limits

1) \(\lim\limits_{x \to 3} \frac{27-x^3}{x-3}\)

2) \(\lim\limits_{x \to 3} \frac{x^2-4x+3}{2x-6}\)

3) \(\lim\limits_{x \to -1} \frac{x^3-1}{x+1}\)

4) \(\lim\limits_{x \to -2} \frac{x+2}{x^5+32}\)

5) \(\lim\limits_{x \to 4} \frac{x^2-2x-8}{x^2-9x+20}\)

6) \(\lim\limits_{x \to -5} \frac{x^3+125}{2x^2-50}\)

7) \(\lim\limits_{x \to -2} \frac{3x^2+5x-2}{4x^2+9x+2}\)

8) \(\lim\limits_{x \to 1} \frac{x^n-1}{x-1}\), n - natural number

9) \(\lim\limits_{x \to 3} \frac{(x-3)(-1)^{[x]}}{x^2-9}\)

10) \(\lim\limits_{x \to 0} \frac{\sqrt[3]{1+mx}-1}{x}\)

11) \(\lim\limits_{x \to 1} \frac{x^n-1}{x-1}\) n - natural number.

12) \(\lim\limits_{x \to 25} \frac{\sqrt{x}-5}{x-25}\)

13) \(\lim\limits_{x \to 0} \frac{\sqrt{x^2+1}-\sqrt{x+1}}{1-\sqrt{x+1}}\)

14) \(\lim\limits_{x \to 0} \frac{\sqrt{x^2+1}-1}{\sqrt{x^2+25}-5}\)

15) \(\lim\limits_{x \to 0} \frac{\sin 3x}{4x}\)

16) \(\lim\limits_{x \to 0} \frac{4x}{3 \sin 2x}\)

17) \(\lim\limits_{x \to +\infty} \frac{\sin x}{x}\)

18) \(\lim\limits_{x \to \pi} \frac{\sin x}{x}\)

19) \(\lim\limits_{x \to \frac{\pi}{2}} \frac{\cos x}{x-\frac{\pi}{2}}\)

20) \(\lim\limits_{x \to 0} \frac{\text{tg } x}{4x}\)

21) \(\lim\limits_{x \to \pi} \frac{8-x}{\sin x}\)

22) \(\lim\limits_{x \to 0} \frac{\sin 2x}{\sin 3x}\)

23) \(\lim\limits_{x \to 0} \frac{\text{tg } 2x}{\text{tg } x}\)

24) \(\lim\limits_{x \to \frac{\pi}{2}} \frac{1+\cos x}{\sin^2 x}\)

25) \(\lim\limits_{x \to \frac{\pi}{4}} \frac{\cos x - \cos \frac{\pi}{4}}{\sin x - \sin \frac{\pi}{4}}\)

26) \(\lim\limits_{x \to 1} \frac{|\text{tg}(x-1)|}{(x-1)^2}\)

27) \(\lim\limits_{x \to 0} \frac{\text{arctg } x}{x}\)

28) \(\lim\limits_{x \to \frac{1}{2}} \frac{\arcsin(1-2x)}{4x^2-1}\)

29) \(\lim\limits_{x \to 0} \frac{\sqrt{1+\sin x}}{x}\)

30) \(\lim\limits_{x \to 0} (1-3x)^{\frac{1}{x}}\)

31) \(\lim\limits_{x \to 0} (1+kx)^{\frac{n}{x}}\)

Ex 2. For the given functions determine if they are continuous at the given points. If not, can they be defined to be continuous.

1) \(f(x) = \frac{x^2-25}{x+5}\) for \(x \neq -5\) and \(f(-5)=-10\).

2) \(f(x) = \frac{\sin x}{x}\) for \(x \neq 0\) and \(f(0)=1\).

3) \(f(x) = \frac{\sin x}{|x|}\) for \(x \neq 0\) and \(f(0)=1\).

4) \(f(x) = x + \frac{1}{x}\)

5) \(f(x) = \frac{x^2-x^3}{|x-1|}\)

6) \(f(x) = x - [x]\)

7) \(f(x) = [x] + [-x]\)

8) \(f(x) = \frac{\sqrt{1+x}-1}{x}\)

9) \(f(x) = x \sin \frac{\pi}{x}\)

10) \(f(x) = \frac{\sin^2 x}{1-\cos x}\)

11) \(x \left[ \frac{1}{x} \right]\) at point \(x=0\).

12) \(x \frac{b}{x} \left[ \frac{x}{a} \right]\) at point \(x=0\).

13) \(\frac{e^{\frac{1}{x}}-1}{e^{\frac{1}{x}}+1}\) at point \(x=0\).

14) \(e^{\frac{1}{1-x^2}}\) at point \(x=1\).

15) \(x e^{\frac{1}{x}}\) at point \(x=0\).

16) \(\frac{x}{2x+e^{\frac{1}{x-1}}}\) at point \(x=1\).