Evaluate 1) lim x-infinity (2x2 - 7)/(6 − 3x2) 2) lim x-3 (x/(x+2) − 3/5)/(x−3)

Question

Evaluate 1) lim x->infinity (2x2 - 7)/(6 − 3x2) 2) lim x->3 (x/(x+2) − 3/5)/(x−3)

ScanSolve AI Tutor · Accepted Answer

Here are the solutions to the limit problems.

:
Evaluate $\lim_{x \to \infty} \frac{2x^2 - 7}{6 - 3x^2}$

Identify the highest power of $x$ in the denominator, which is $x^2$ .
Divide every term in the numerator and denominator by $x^2$ .
$\lim_{x \to \infty} \frac{\frac{2x^2}{x^2} - \frac{7}{x^2}}{\frac{6}{x^2} - \frac{3x^2}{x^2}}$
Simplify the expression.
$\lim_{x \to \infty} \frac{2 - \frac{7}{x^2}}{\frac{6}{x^2} - 3}$
Apply the limit property that $\lim_{x \to \infty} \frac{c}{x^n} = 0$ for any constant $c$ and $n > 0$ .
$\frac{2 - 0}{0 - 3}$
Calculate the final value.
$\frac{2}{-3} = -\frac{2}{3}$
The limit is $\boxed{-\frac{2}{3}}$ .

:
Evaluate $\lim_{x \to 3} \frac{\frac{x}{x+2} - \frac{3}{5}}{x-3}$
Combine the fractions in the numerator by finding a common denominator.
$\frac{x}{x+2} - \frac{3}{5} = \frac{5x - 3(x+2)}{5(x+2)}$
Simplify the numerator.
$\frac{5x - 3x - 6}{5(x+2)} = \frac{2x - 6}{5(x+2)}$
Substitute the simplified numerator back into the limit expression and rewrite it as a single fraction.
$\lim_{x \to 3} \frac{\frac{2x - 6}{5(x+2)}}{x-3} = \lim_{x \to 3} \frac{2x - 6}{5(x+2)(x-3)}$
Factor the numerator to find a common term with the denominator.
$\lim_{x \to 3} \frac{2(x - 3)}{5(x+2)(x-3)}$
Cancel out the $(x-3)$ term, since $x \neq 3$ as $x$ approaches 3.
$\lim_{x \to 3} \frac{2}{5(x+2)}$
Substitute $x=3$ into the simplified expression.
$\frac{2}{5(3+2)} = \frac{2}{5(5)} = \frac{2}{25}$
The limit is $\boxed{\frac{2}{25}}$ .

Evaluate 1) lim x->infinity (2x2 - 7)/(6 - 3x2) 2) lim x->3 (x/(x+2) - 3/5)/(x-3)